FILING PURSUANT TO RULE 424(b)(2)
                                           REGISTRATION STATEMENT NO. 333-115195


                           PROSPECTUS SUPPLEMENT NO. 1
                        (TO PROSPECTUS DATED MAY 5, 2004)

                                GERON CORPORATION

                                  COMMON STOCK,
                      WARRANTS TO PURCHASE COMMON STOCK AND
                 COMMON STOCK ISSUABLE UPON EXERCISE OF WARRANTS

     You should read this prospectus supplement and the accompanying prospectus
carefully before you invest. Both documents contain information you should
consider carefully before making your investment decision.

     We are offering pursuant to the terms and conditions of securities purchase
agreements an aggregate of 4,859,016 shares of our common stock, par value
$0.001 per share, warrants to purchase an aggregate of 6,042,623 additional
shares of common stock and the shares of our common stock to be issued upon
exercise of such warrants. We will receive immediate proceeds from the sale of
these shares and warrants equal to approximately $40,000,000, less our expenses
relating to the sale, which are estimated to be $120,000.

     We are issuing three types of warrants, to which we refer as the A
Warrants, the B Warrants and the C Warrants, respectively. The A Warrants are
warrants to purchase up to an aggregate of 2,295,082 shares of common stock,
which are exercisable from time to time at a price of $8.62 per share during the
period from May 11, 2005 until November 12, 2008. The B Warrants are warrants to
purchase up to an aggregate of 2,049,180 shares of common stock, which are
exercisable from time to time at a price of $6.10 per share during the period
from November 12, 2004 until January 11, 2005. The terms of the B Warrants
provide that the sixty-day term for exercise of the warrants may, in certain
circumstances, be extended until November 12, 2007, upon payment by the warrant
holder of an extension fee equal to $6.09 per warrant share, such that if
extended with respect to all warrant shares, the aggregate extension fee payable
to the Company would be approximately $12.5 million. The C Warrants, for which
we will receive proceeds per underlying share equal to the purchase price per
share of Common Stock ($6.10), are warrants to purchase up to an aggregate of
1,698,361 shares of Common Stock which are exercisable from time to time for
nominal additional consideration during the period from November 12, 2004 until
November 12, 2006.

     None of the A Warrants, B Warrants or C Warrants is exercisable to the
extent that exercise of such warrants would cause any purchaser and its
affiliates to beneficially own more than 4.9% of the outstanding shares of the
Company's Common Stock. We will receive additional proceeds of approximately
$33.3 million (less our expenses relating to the sales, which are estimated to
be $15,000) upon the full exercise of the A Warrants, the B Warrants and the C
Warrants (the portion of the proceeds received upon exercise of the C Warrants
will be nominal).

     There will be no warrant agent in connection with the issuance of the A
Warrants, the B Warrants or the C Warrants.

                LIMITATION ON THE USE OF OUR NET OPERATING LOSSES

We have significant amounts of net operating loss ("NOL") carry forwards
(including, as of December 31, 2003, domestic federal NOL carry forwards of
approximately $188,000,000, which will expire at various dates beginning 2006
through 2023, if not utilized). These NOLs may be used to offset future taxable
income, to the extent we generate any taxable income, and thereby reduce or
eliminate our future federal income taxes otherwise payable. However, the
Internal Revenue Code of 1986, as amended, imposes significant limitations on
the utilization of NOLs in the event of an "ownership change," as defined in
Section 382 of the Code. This Section 382 limitation is an annual limitation on
the amount of pre-ownership change NOLs that a corporation may use to offset its
post-ownership change income. We may or may not have experienced an ownership
change as a result of events in the past and/or the sale of shares of common
stock and warrants pursuant to this prospectus supplement (or a combination
thereof). If so, the use of our NOLs (or a portion thereof) against our future
taxable income may be subject to a limitation under Section 382.



                    MATERIAL U.S. FEDERAL TAX CONSIDERATIONS

     The following is a general discussion of the material U.S. federal income
tax consequences of the ownership and disposition of our common stock and
Warrants, but is not a complete analysis of all the potential tax consequences
relating thereto. For the purposes of this discussion, a U.S. Holder is any
beneficial owner of our common stock or Warrants that is treated for U.S.
federal income tax purposes as:

     o    an individual citizen or resident of the United States;

     o    a corporation (or other entity taxable as a corporation) created or
          organized in the United States or under the laws of the United States
          or any political subdivision thereof;

     o    an estate whose income is subject to U.S. federal income tax
          regardless of its source; or

     o    a trust (x) if a court within the United States is able to exercise
          primary supervision over the administration of the trust and one or
          more United States persons have the authority to control all
          substantial decisions of the trust or (y) which has made a valid
          election to be treated as a United States person.

     If a partnership holds our common stock or Warrants, the tax treatment of a
partner will generally depend on the status of the partner and upon the
activities of the partnership. Accordingly, partnerships which hold our common
stock and partners in such partnerships should consult their own tax advisors.

     This discussion does not address all aspects of U.S. federal income
taxation that may be relevant in light of a U.S. holder's special tax status or
special circumstances. U.S. expatriates, insurance companies, tax-exempt
organizations, dealers in securities, insurance companies, banks or other
financial institutions, "controlled foreign corporations," "passive foreign
investment companies," "foreign personal holding companies," corporations that
accumulate earnings to avoid U.S. federal income tax, U.S. expatriates, U.S.
Holders whose functional currency is not the U.S. dollar, tax-exempt
organizations, and investors that hold our common stock as part of a hedge,
straddle or conversion transaction are among those categories of potential
investors that are subject to special rules not covered in this discussion. This
discussion does not address any tax consequences arising under the laws of any
state, local or non-U.S. taxing jurisdiction. Furthermore, the following
discussion is based on current provisions of the Internal Revenue Code of 1986,
as amended, and Treasury Regulations and administrative and judicial
interpretations thereof, all as in effect on the date hereof, and all of which
are subject to change, possibly with retroactive effect. Accordingly, each
Non-U.S. Holder should consult its own tax advisors regarding the U.S. federal,
state, local and non-U.S. income and other tax consequences of acquiring,
holding and disposing of shares of our common stock.



U.S. HOLDERS

Dividends Paid on Common Stock

     A U.S. Holder generally will be required to include in gross income as
ordinary dividend income the amount of any distributions paid on the common
stock to the extent that such distributions are paid out of our current or
accumulated earnings and profits as determined for U.S. federal income tax
purposes. Distributions in excess of such earnings and profits will reduce the
U.S. Holder's tax basis in its common stock and, to the extent such excess
distributions exceed such tax basis, will be treated as gain from a sale or
exchange of such common stock. Corporate holders may be entitled to a dividends
received deduction with respect to such distributions and are urged to consult
their own tax advisors in this regard. With respect to non-corporate U.S.
Holders for taxable years beginning before January 1, 2009, dividends may be
taxed at the lower applicable capital gains rate provided that certain holding
period requirements are met.

Disposition of Common Stock

     Upon the sale or other disposition of common stock, a U.S. Holder generally
will recognize capital gain or loss equal to the difference between the amount
realized on the sale and such holder's adjusted tax basis in the common stock. A
U.S. Holder's tax basis in its common stock will equal that portion of the cost
of the Unit that is allocated to the common stock (based on the relative fair
market values of the common stock and the Warrant comprising the Unit). Gain or
loss upon the disposition of the common stock will be a long-term if, at the
time of the disposition, the holding period for the common stock exceeds on
year.

Tax Treatment of Warrants

     A U.S. Holder will recognize gain or loss upon the sale, redemption, lapse
or other taxable disposition of a Warrant in an amount equal to the difference
between the amount of cash and the fair market value of other property received
(if any) by the U.S. Holder and the U.S. Holder's tax basis in the Warrant. A
U.S. Holder's tax basis in a Warrant will equal that portion of the cost of the
Unit that is allocated to the Warrant (based on the relative fair market values
of the common stock and the Warrant comprising the Unit). Such gain or loss will
be capital gain or loss if the common stock to which the Warrants relate would
be a capital asset in the hands of the Warrant holder and will be long-term
capital gain or loss if the holding period exceeds one year.

     The cash exercise of a Warrant will not, and the cashless exercise of a
Warrant should not, be a taxable event for the exercising U.S. Holder, except
with respect to cash, if any, received in lieu of a fractional share. The
Internal Revenue Service (the "IRS") may argue, however, that the surrender of
one or more Warrants in payment of the exercise price of another Warrant upon a
cashless exercise results in taxable gain or loss to the exercising U.S. Holder
in an amount equal to the difference between the exercise price deemed paid and
the tax basis in the shares of common stock received upon exercise of a Warrant
equal to such U.S. Holder's tax basis in the Warrant surrendered, plus any gain
recognized upon a cashless exercise or the amount of cash paid for the exercise
price, as adjusted for a fractional share for which cash is received, if any. A
U.S. Holder will generally have a holding period in shares of common stock
acquired upon exercise of a Warrant that commences on the day after the date of
exercise of the Warrant. With respect to the cashless exercise of a Warrant, it
is possible that a U.S. Holder may have holding period in shares of common stock
received in exchange for the surrender of one or more Warrants which includes
the holding period of the Warrants so surrendered.

     If the exercise price is nominal, it is possible that the Warrants may be
deemed to have been exercised for tax purposes on the date on which they first
became exercisable or possibly on the date issued, regardless of whether they
are actually exercised on the date on which they are first exercisable. As a
result, it is possible that the holding period of shares of common stock may be
deemed to have begun on the date on which the Warrants first became exercisable
or possibly on the date issued.



     An adjustment to the exercise price or conversion ratio of the Warrants, or
the failure to make such adjustments, may in certain circumstances result in
constructive distributions to the holders of the Warrants that could be taxable
as dividends under Section 305 of the Code. In such event, a holder's tax basis
in the Warrant would be increased by the amount of any such dividend.

NON-U.S. HOLDERS

     A Non-U.S. Holder is a beneficial owner of our common stock or Warrants
that is not a U.S. Holder or a U.S. domestic partnership or other entity treated
as a U.S. domestic partnership for U.S. federal income tax purposes.

Dividends Paid on Common Stock

     Payments on our common stock will constitute dividends for U.S. federal
income tax purposes to the extent paid from our current or accumulated earnings
and profits, as determined under U.S. federal income tax principles. Amounts not
treated as dividends for U.S. federal income tax purposes will constitute a
return of capital and will first be applied against and reduce a holder's
adjusted basis in the common stock, but not below zero, and then the excess, if
any, will be treated as gain from the sale of the common stock (the tax
treatment of which is described below).

     Amounts treated as dividends paid to a Non-U.S. Holder of common stock
generally will be subject to U.S. withholding tax either at a rate of 30% of the
gross amount of the dividends or such lower rate as may be specified by an
applicable tax treaty. In order to receive a reduced treaty rate, a Non-U.S.
Holder must provide a valid IRS Form W-8BEN or other successor form certifying
qualification for the reduced rate.

     Dividends received by a Non-U.S. Holder that are effectively connected with
a United States trade or business conducted by the Non-U.S. Holder are exempt
from such withholding tax. In order to obtain this exemption, a Non-U.S. Holder
must provide a valid IRS Form W-8ECI or other successor form properly certifying
such exemption. Such effectively connected dividends, although not subject to
withholding tax, are generally taxed at the same graduated rates applicable to
United States persons, net of allowable deductions and credits.

     In addition to the graduated tax described above, dividends received by a
corporate Non-U.S. Holder that are effectively connected with a United States
trade or business of such holder may also be subject to a branch profits tax at
a rate of 30% or such lower rate as may be specified by an applicable tax
treaty.

     A Non-U.S. Holder may obtain a refund of any excess amounts currently
withheld if an appropriate claim for refund is filed timely with the IRS. If a
Non-U.S. Holder holds our common stock through a foreign partnership or a
foreign intermediary, the foreign partnership or foreign intermediary will also
be required to comply with additional certification requirements.

Gain on Disposition of Common Stock or Warrants

     A Non-U.S. Holder generally will not be subject to U.S. federal income tax
on any gain realized upon the sale or other disposition of our common stock or
Warrants unless:



     o    the gain is effectively connected with a United States trade or
          business of the Non-U.S. Holder or, if a tax treaty applies, as
          attributable to a United States permanent establishment maintained by
          such non-United States holder;

     o    the Non-U.S. Holder is an individual who holds his or her common stock
          or Warrants as a capital asset (generally, an asset held for
          investment purposes) and who is present in the United States for a
          period or periods aggregating 183 days or more during the taxable year
          in which the sale or disposition occurs and other conditions are met;
          or

     o    our common stock or Warrants constitute a United States real property
          interest by reason of our status as a "United States real property
          holding corporation" (a "USRPHC") for U.S. federal income tax purposes
          at any time within the shorter of the five-year period preceding the
          disposition or the holder's holding period for our common stock.

     We believe that we are not currently and do not expect to become a USRPHC.
However, because the determination of whether we are a USRPHC depends on the
fair market value of our United States real property interests relative to the
fair market value of our other business assets, there can be no assurance that
we will not become a USRPHC in the future.

     Unless an applicable treaty provides otherwise, gain described in the first
bullet point above will be subject to U.S. federal income tax imposed on net
income on the same basis that applies to United States persons generally and,
for corporate holders under certain circumstances, the branch profits tax, but
will generally not be subject to withholding. Non-U.S. Holders should consult
any applicable income tax treaties that may provide for different rules.



     Our common stock is quoted on the Nasdaq National Market under the symbol
"GERN." On November 11, 2004, the last reported sale price of our common stock
on the Nasdaq National Market was $6.96 per share. As of November 11, 2004, we
had 45,637,319 shares of common stock outstanding.

     Investing in our common stock involves certain risks. See "Risk Factors"
beginning on page 1 of the prospectus, including the specific risks set forth
under the caption "Additional Factors That May Affect Future Results" and
incorporated by reference from the Company's filings made with the Securities
and Exchange Commission under the Securities Exchange of Act, as amended,
between the date of the prospectus and the termination of the offering.

     You should rely on the information provided or incorporated by reference in
this prospectus supplement and the prospectus. We have not authorized anyone
else to provide you with different information. You should not assume that the
information in this prospectus supplement is accurate as of any date other than
the date on the front of these documents.

     Neither the U.S. Securities and Exchange Commission nor any state
securities commission has approved or disapproved of the securities or passed
upon the accuracy or adequacy of this prospectus. Any representation to the
contrary is a criminal offense.

     The date of this prospectus supplement is November 12, 2004.

                 TABLE OF CONTENTS

                    PROSPECTUS
                                                                               
About this Prospectus                                                          1
About Geron                                                                    1
Risk Factors                                                                   1
Forward-Looking Statements                                                     2
Ratio of Earnings to Fixed Charges                                             2
Use of Proceeds                                                                2
Plan of Distribution                                                           2
Description of Debt Securities                                                 4
Description of Common Stock                                                   12
Description of Preferred Stock                                                13
Description of Warrants                                                       15
Certain Provisions of Delaware Law and of the Company's Charter and Bylaws    16
Legal Matters                                                                 17
Experts                                                                       17
Limitation on Liability and Disclosure of Commission Position on
Indemnification for Securities Act Liabilities                                17
Where You Can Find More Information                                           18