<PAGE>

                                                                     Exhibit 4.5


                              GUARANTEE AGREEMENT


                              GREATER BAY BANCORP


                          Dated as of March 23, 2000
<PAGE>

                               Table of Contents

<TABLE>
<CAPTION>
                                                                               Page
<S>                                                                            <C>
                                   ARTICLE I
                        DEFINITIONS AND INTERPRETATION

SECTION 1.1  Definitions and Interpretation...................................  1

                                  ARTICLE II
                         POWERS, DUTIES AND RIGHTS OF
                               GUARANTEE TRUSTEE

SECTION 2.1  Powers and Duties of the Guarantee Trustee.......................  4
SECTION 2.2  Certain Rights of Guarantee Trustee..............................  5
SECTION 2.3  Not Responsible for Recitals or Issuance of Guarantee............  7
SECTION 2.4  Events of Default; Waiver........................................  8
SECTION 2.5  Events of Default; Notice........................................  8

                                  ARTICLE III
                               GUARANTEE TRUSTEE

SECTION 3.1  Guarantee Trustee; Eligibility...................................  8
SECTION 3.2  Appointment, Removal and Resignation of Guarantee Trustee........  9

                                  ARTICLE IV
                                   GUARANTEE

SECTION 4.1  Guarantee.......................................................  10
SECTION 4.2  Waiver of Notice and Demand.....................................  10
SECTION 4.3  Obligations Not Affected........................................  10
SECTION 4.4  Rights of Holders...............................................  11
SECTION 4.5  Guarantee of Payment............................................  11
SECTION 4.6  Subrogation.....................................................  12
SECTION 4.7  Independent Obligations.........................................  12
SECTION 4.8  Enforcement by a Beneficiary....................................  12

                                   ARTICLE V
                   LIMITATION OF TRANSACTIONS; SUBORDINATION

SECTION 5.1  Limitation of Transactions......................................  12
SECTION 5.2  Ranking.........................................................  13

                                  ARTICLE VI
                                  TERMINATION

SECTION 6.1  Termination.....................................................  13

                                  ARTICLE VII
                                INDEMNIFICATION

SECTION 7.1  Exculpation.....................................................  14
SECTION 7.2  Indemnification.................................................  14
SECTION 7.3  Compensation; Reimbursement of Expenses.........................  15
</TABLE>

                                      i
<PAGE>

<TABLE>
<S>                                                                            <C>
                                 ARTICLE VIII
                                 MISCELLANEOUS

SECTION 8.1  Successors and Assigns..........................................  16
SECTION 8.2  Amendments......................................................  16
SECTION 8.3  Notices.........................................................  16
SECTION 8.4  Benefit.........................................................  17
SECTION 8.5  Governing Law...................................................  17
</TABLE>

                                      ii
<PAGE>

                              GUARANTEE AGREEMENT
                              -------------------

          This GUARANTEE AGREEMENT (the "Guarantee"), dated as of March 23,
2000, is executed and delivered by Greater Bay Bancorp, a bank holding company
incorporated in California (the "Guarantor"), and The Bank of New York, as
trustee (the "Guarantee Trustee"), for the benefit of the Holders (as defined
herein) from time to time of the Capital Securities (as defined herein) of GBB
Capital III, a Delaware statutory business trust (the "Issuer").

          WHEREAS, pursuant to an Amended and Restated Declaration of Trust (the
"Declaration"), dated as of March 23, 2000, among the trustees named therein of
the Issuer, Greater Bay Bancorp, as sponsor, and the holders from time to time
of undivided beneficial interests in the assets of the Issuer, the Issuer is
issuing on the date hereof securities, having an aggregate liquidation amount of
up to $9,500,000, designated the Fixed Rate Capital Trust Pass-through
Securities/(R)/ (the "Capital Securities");

          WHEREAS, as incentive for the Holders to purchase the Capital
Securities, the Guarantor desires irrevocably and unconditionally to agree, to
the extent set forth in this Guarantee, to pay to the Holders of Capital
Securities the Guarantee Payments (as defined herein) and to make certain other
payments on the terms and conditions set forth herein; and

          NOW, THEREFORE, in consideration of the purchase by each Holder of the
Capital Securities, which purchase the Guarantor hereby agrees shall benefit the
Guarantor, the Guarantor executes and delivers this Guarantee for the benefit of
the Holders.

                                   ARTICLE I
                        DEFINITIONS AND INTERPRETATION

SECTION 1.1  Definitions and Interpretation
             ------------------------------

          In this Guarantee, unless the context otherwise requires:

          (a)  capitalized terms used in this Guarantee but not defined in the
     preamble above have the respective meanings assigned to them in this
     Section 1.1;

          (b)  a term defined anywhere in this Guarantee has the same meaning
     throughout;

          (c)  all references to "the Guarantee" or "this Guarantee" are to this
     Guarantee as modified, supplemented or amended from time to time;

          (d)  all references in this Guarantee to Articles and Sections are to
     Articles and Sections of this Guarantee, unless otherwise specified;

          (e)  terms defined in the Declaration as at the date of execution of
     this Guarantee have the same meanings when used in this Guarantee, unless
     otherwise defined in this Guarantee or unless the context otherwise
     requires; and

          (f)  a reference to the singular includes the plural and vice versa.
<PAGE>

          "Beneficiaries" means any Person to whom the Trust is or hereafter
becomes indebted or liable.

          "Corporate Trust Office" means the office of the Guarantee Trustee at
which the corporate trust business of the Guarantee Trustee shall, at any
particular time, be principally administered, which office at the date of
execution of this Guarantee Agreement is located at 101 Barclay Street, Floor
21W, New York, NY 10286.

          "Covered Person" means any Holder of Capital Securities.

          "Debentures" means the junior subordinated debentures of Greater Bay
Bancorp, designated the 10 7/8% Junior Subordinated Deferrable Interest
Debentures due 2030, held by the Institutional Trustee (as defined in the
Declaration) of the Issuer.

          "Declaration Event of Default" means an "Event of Default" as defined
in the Declaration.

          "Event of Default" has the meaning set forth in Section 2.4.

          "Guarantee Payments" means the following payments or distributions,
without duplication, with respect to the Capital Securities, to the extent not
paid or made by the Issuer: (i) any accrued and unpaid Distributions (as defined
in the Declaration) which are required to be paid on such Capital Securities to
the extent the Issuer shall have funds available therefor, (ii) the Redemption
Price (as defined in the Indenture) to the extent the Issuer has funds available
therefor, with respect to any Capital Securities called for redemption by the
Issuer, (iii) the Special Redemption Price (as defined in the Indenture) to the
extent the Issuer has funds available therefor, with respect to Capital
Securities redeemed upon the occurrence of a Special Event (as defined in the
Indenture), and (iv) upon a voluntary or involuntary liquidation, dissolution,
winding-up or termination of the Issuer (other than in connection with the
distribution of Debentures to the Holders of the Capital Securities in exchange
therefor as provided in the Declaration), the lesser of (a) the aggregate of the
liquidation amount and all accrued and unpaid Distributions on the Capital
Securities to the date of payment, to the extent the Issuer shall have funds
available therefor, and (b) the amount of assets of the Issuer remaining
available for distribution to Holders in liquidation of the Issuer after
satisfaction of liabilities to creditors of the Issuer as required by applicable
law (in either case, the "Liquidation Distribution").

          "Guarantee Trustee" means The Bank of New York, until a Successor
Guarantee Trustee has been appointed and has accepted such appointment pursuant
to the terms of this Guarantee and thereafter means each such Successor
Guarantee Trustee.

          "Holder" shall mean any holder, as registered on the books and records
of the Issuer, of any Capital Securities; provided, however, that, in
                                          --------  -------
determining whether the holders of the requisite percentage of Capital
Securities have given any request, notice, consent or waiver hereunder, "Holder"
shall not include the Guarantor or any Affiliate of the Guarantor.

                                       2
<PAGE>

          "Indemnified Person" means the Guarantee Trustee, any Affiliate of the
Guarantee Trustee, or any officers, directors, shareholders, members, partners,
employees, representatives, nominees, custodians or agents of the Guarantee
Trustee.

          "Indenture" means the Indenture dated as of March 23,  2000, between
the Guarantor and The Bank of New York, not in its individual capacity but
solely as trustee, and any indenture supplemental thereto pursuant to which the
Debentures are to be issued to the Institutional Trustee of the Issuer.

          "Liquidation Distribution" has the meaning set forth in the definition
of "Guarantee Payments" herein.

          "Majority in liquidation amount of the Capital Securities" means
Holder(s) of outstanding Capital Securities, voting together as a class, but
separately from the holders of Common Securities, of more than 50% of the
aggregate liquidation amount (including the stated amount that would be paid on
redemption, liquidation or otherwise, plus accrued and unpaid Distributions to
the date upon which the voting percentages are determined) of all Capital
Securities then outstanding.

          "Obligations" means any costs, expenses or liabilities (but not
including liabilities related to taxes) of the Trust, other than obligations of
the Trust to pay to holders of any Trust Securities the amounts due such holders
pursuant to the terms of the Trust Securities.

          "Officer's Certificate" means, with respect to any Person, a
certificate signed by one Authorized Officer of such Person.  Any Officer's
Certificate delivered with respect to compliance with a condition or covenant
provided for in this Guarantee shall include:

          (a)  a statement that each officer signing the Officer's Certificate
     has read the covenant or condition and the definitions relating thereto;

          (b)  a brief statement of the nature and scope of the examination or
     investigation undertaken by each officer in rendering the Officer's
     Certificate;

          (c)  a statement that each such officer has made such examination or
     investigation as, in such officer's opinion, is necessary to enable such
     officer to express an informed opinion as to whether or not such covenant
     or condition has been complied with; and

          (d)  a statement as to whether, in the opinion of each such officer,
     such condition or covenant has been complied with.

          "Other Debentures" shall mean all junior subordinated debentures other
than the Debentures issued by the Company from time to time and sold to trusts
other than the Issuer (if any), in each case similar to the Issuer.

          "Other Guarantees" shall mean all guarantees other than this Guarantee
issued by the Company with respect to preferred beneficial interests (if any)
issued to trusts other than the Issuer (if any), in each case similar to the
Issuer.

                                       3
<PAGE>

          "Person" means a legal person, including any individual, corporation,
estate, partnership, joint venture, association, joint stock company, limited
liability company, trust, unincorporated association, or government or any
agency or political subdivision thereof, or any other entity of whatever nature.

          "Redemption Price" has the meaning set forth in the definition of
"Guarantee Payments" herein.

          "Responsible Officer" means, with respect to the Guarantee Trustee,
any officer within the Corporate Trust Office of the Guarantee Trustee,
including any Vice President, Assistant Vice President, Secretary, Assistant
Secretary or any other officer of the Guarantee Trustee customarily performing
functions similar to those performed by any of the above designated officers and
also, with respect to a particular corporate trust matter, any other officer to
whom such matter is referred because of that officer's knowledge of and
familiarity with the particular subject.

          "Successor Guarantee Trustee" means a successor Guarantee Trustee
possessing the qualifications to act as Guarantee Trustee under Section 3.1.

          "Trust Securities" means the Common Securities and the Capital
Securities.

                                  ARTICLE II
                         POWERS, DUTIES AND RIGHTS OF
                               GUARANTEE TRUSTEE

SECTION 2.1  Powers and Duties of the Guarantee Trustee
             ------------------------------------------

          (a)  This Guarantee shall be held by the Guarantee Trustee for the
     benefit of the Holders of the Capital Securities, and the Guarantee Trustee
     shall not transfer this Guarantee to any Person except a Holder of Capital
     Securities exercising his or her rights pursuant to Section 4.4(b) or to a
     Successor Guarantee Trustee on acceptance by such Successor Guarantee
     Trustee of its appointment to act as Successor Guarantee Trustee.  The
     right, title and interest of the Guarantee Trustee shall automatically vest
     in any Successor Guarantee Trustee, and such vesting and cessation of title
     shall be effective whether or not conveyancing documents have been executed
     and delivered pursuant to the appointment of such Successor Guarantee
     Trustee.

          (b)  If an Event of Default actually known to a Responsible Officer of
     the Guarantee Trustee has occurred and is continuing, the Guarantee Trustee
     shall enforce this Guarantee for the benefit of the Holders of the Capital
     Securities.

          (c)  The Guarantee Trustee, before the occurrence of any Event of
     Default and after curing all Events of Default that may have occurred,
     shall undertake to perform only such duties as are specifically set forth
     in this Guarantee, and no implied covenants shall be read into this
     Guarantee against the Guarantee Trustee.  In case an Event of Default has
     occurred (that has not been cured or waived pursuant to Section 2.4) and is
     actually known to a Responsible Officer of the Guarantee Trustee, the
     Guarantee Trustee shall exercise such of the rights and powers vested in it
     by this Guarantee, and use the same

                                       4
<PAGE>

     degree of care and skill in its exercise thereof, as a prudent person would
     exercise or use under the circumstances in the conduct of his or her own
     affairs.

           (d) No provision of this Guarantee shall be construed to relieve the
     Guarantee Trustee from liability for its own negligent action, its own
     negligent failure to act, or its own willful misconduct, except that:

                  (i)    prior to the occurrence of any Event of Default and
          after the curing or waiving of all such Events of Default that may
          have occurred:

                             (A)  the duties and obligations of the Guarantee
               Trustee shall be determined solely by the express provisions of
               this Guarantee, and the Guarantee Trustee shall not be liable
               except for the performance of such duties and obligations as are
               specifically set forth in this Guarantee, and no implied
               covenants or obligations shall be read into this Guarantee
               against the Guarantee Trustee; and

                            (B)  in the absence of bad faith on the part of the
               Guarantee Trustee, the Guarantee Trustee may conclusively rely,
               as to the truth of the statements and the correctness of the
               opinions expressed therein, upon any certificates or opinions
               furnished to the Guarantee Trustee and conforming to the
               requirements of this Guarantee; but in the case of any such
               certificates or opinions furnished to the Guarantee Trustee, the
               Guarantee Trustee shall be under a duty to examine the same to
               determine whether or not they conform to the requirements of this
               Guarantee;

                  (ii)   the Guarantee Trustee shall not be liable for any error
          of judgment made in good faith by a Responsible Officer of the
          Guarantee Trustee, unless it shall be proved that such Responsible
          Officer of the Guarantee Trustee or the Guarantee Trustee was
          negligent in ascertaining the pertinent facts upon which such judgment
          was made;

                  (iii)  the Guarantee Trustee shall not be liable with respect
          to any action taken or omitted to be taken by it in good faith in
          accordance with the direction of the Holders of not less than a
          Majority in liquidation amount of the Capital Securities relating to
          the time, method and place of conducting any proceeding for any remedy
          available to the Guarantee Trustee, or exercising any trust or power
          conferred upon the Guarantee Trustee under this Guarantee; and

                  (iv)   no provision of this Guarantee shall require the
          Guarantee Trustee to expend or risk its own funds or otherwise incur
          personal financial liability in the performance of any of its duties
          or in the exercise of any of its rights or powers, if the Guarantee
          Trustee shall have reasonable grounds for believing that the repayment
          of such funds is not reasonably assured to it under the terms of this
          Guarantee, or security and indemnity, reasonably satisfactory to the
          Guarantee Trustee, against such risk or liability is not reasonably
          assured to it.

SECTION 2.2  Certain Rights of Guarantee Trustee
             -----------------------------------

                                       5
<PAGE>

          (a)  Subject to the provisions of Section 2.1:

                   (i)    The Guarantee Trustee may conclusively rely, and shall
          be fully protected in acting or refraining from acting upon, any
          resolution, certificate, statement, instrument, opinion, report,
          notice, request, direction, consent, order, bond, debenture, note,
          other evidence of indebtedness or other paper or document believed by
          it to be genuine and to have been signed, sent or presented by the
          proper party or parties.

                   (ii)   Any direction or act of the Guarantor contemplated by
          this Guarantee shall be sufficiently evidenced by an Officer's
          Certificate.

                   (iii)  Whenever, in the administration of this Guarantee, the
          Guarantee Trustee shall deem it desirable that a matter be proved or
          established before taking, suffering or omitting any action hereunder,
          the Guarantee Trustee (unless other evidence is herein specifically
          prescribed) may, in the absence of bad faith on its part, request and
          conclusively rely upon an Officer's Certificate of the Guarantor
          which, upon receipt of such request, shall be promptly delivered by
          the Guarantor.

                   (iv)   The Guarantee Trustee shall have no duty to see to any
          recording, filing or registration of any instrument (or any
          rerecording, refiling or registration thereof).

                   (v)    The Guarantee Trustee may consult with counsel of its
          selection, and the advice or opinion of such counsel with respect to
          legal matters shall be full and complete authorization and protection
          in respect of any action taken, suffered or omitted by it hereunder in
          good faith and in accordance with such advice or opinion. Such counsel
          may be counsel to the Guarantor or any of its Affiliates and may
          include any of its employees. The Guarantee Trustee shall have the
          right at any time to seek instructions concerning the administration
          of this Guarantee from any court of competent jurisdiction.

                   (vi)   The Guarantee Trustee shall be under no obligation to
          exercise any of the rights or powers vested in it by this Guarantee at
          the request or direction of any Holder, unless such Holder shall have
          provided to the Guarantee Trustee such security and indemnity,
          reasonably satisfactory to the Guarantee Trustee, against the costs,
          expenses (including attorneys' fees and expenses and the expenses of
          the Guarantee Trustee's agents, nominees or custodians) and
          liabilities that might be incurred by it in complying with such
          request or direction, including such reasonable advances as may be
          requested by the Guarantee Trustee; provided, however, that nothing
                                              --------  -------
          contained in this Section 2.2(a)(vi) shall be taken to relieve the
          Guarantee Trustee, upon the occurrence of an Event of Default, of its
          obligation to exercise the rights and powers vested in it by this
          Guarantee.


                   (vii)  The Guarantee Trustee shall not be bound to make any
          investigation into the facts or matters stated in any resolution,
          certificate, statement, instrument,

                                       6
<PAGE>

          opinion, report, notice, request, direction, consent, order, bond,
          debenture, note, other evidence of indebtedness or other paper or
          document, but the Guarantee Trustee, in its discretion, may make such
          further inquiry or investigation into such facts or matters as it may
          see fit.

                  (viii)  The Guarantee Trustee may execute any of the trusts or
          powers hereunder or perform any duties hereunder either directly or by
          or through agents, nominees, custodians or attorneys, and the
          Guarantee Trustee shall not be responsible for any misconduct or
          negligence on the part of any agent or attorney appointed with due
          care by it hereunder.

                  (ix)    Any action taken by the Guarantee Trustee or its
          agents hereunder shall bind the Holders of the Capital Securities, and
          the signature of the Guarantee Trustee or its agents alone shall be
          sufficient and effective to perform any such action. No third party
          shall be required to inquire as to the authority of the Guarantee
          Trustee to so act or as to its compliance with any of the terms and
          provisions of this Guarantee, both of which shall be conclusively
          evidenced by the Guarantee Trustee's or its agent's taking such
          action.

                  (x)     Whenever in the administration of this Guarantee the
          Guarantee Trustee shall deem it desirable to receive instructions with
          respect to enforcing any remedy or right or taking any other action
          hereunder, the Guarantee Trustee (A) may request instructions from the
          Holders of a Majority in liquidation amount of the Capital Securities,
          (B) may refrain from enforcing such remedy or right or taking such
          other action until such instructions are received, and (C) shall be
          protected in conclusively relying on or acting in accordance with such
          instructions.

                  (xi)    The Guarantee Trustee shall not be liable for any
          action taken, suffered, or omitted to be taken by it in good faith and
          reasonably believed by it to be authorized or within the discretion or
          rights or powers conferred upon it by this Guarantee.

          (b)  No provision of this Guarantee shall be deemed to impose any duty
     or obligation on the Guarantee Trustee to perform any act or acts or
     exercise any right, power, duty or obligation conferred or imposed on it,
     in any jurisdiction in which it shall be illegal or in which the Guarantee
     Trustee shall be unqualified or incompetent in accordance with applicable
     law to perform any such act or acts or to exercise any such right, power,
     duty or obligation.  No permissive power or authority available to the
     Guarantee Trustee shall be construed to be a duty.

SECTION 2.3  Not Responsible for Recitals or Issuance of Guarantee
             -----------------------------------------------------

          The recitals contained in this Guarantee shall be taken as the
statements of the Guarantor, and the Guarantee Trustee does not assume any
responsibility for their correctness.  The Guarantee Trustee makes no
representation as to the validity or sufficiency of this Guarantee.

                                       7
<PAGE>

SECTION 2.4  Events of Default; Waiver
             -------------------------

          (a)  An Event of Default under this Guarantee will occur upon the
failure of the Guarantor to perform any of its payment or other obligations
hereunder.

          (b)  The Holders of a Majority in liquidation amount of Capital
Securities may, voting or consenting as a class, on behalf of the Holders of all
of the Capital Securities, waive any past Event of Default and its consequences.
Upon such waiver, any such Event of Default shall cease to exist, and shall be
deemed to have been cured, for every purpose of this Guarantee, but no such
waiver shall extend to any subsequent or other default or Event of Default or
impair any right consequent thereon.

SECTION 2.5  Events of Default; Notice
             -------------------------

          (a)  The Guarantee Trustee shall, within 90 days after the occurrence
     of an Event of Default, transmit by mail, first class postage prepaid, to
     the Holders of the Capital Securities, notices of all Events of Default
     actually known to a Responsible Officer of the Guarantee Trustee, unless
     such defaults have been cured before the giving of such notice, provided,
                                                                     --------
     however, that the Guarantee Trustee shall be protected in withholding such
     -------
     notice if and so long as a Responsible Officer of the Guarantee Trustee in
     good faith determines that the withholding of such notice is in the
     interests of the Holders of the Capital Securities.

          (b)  The Guarantee Trustee shall not be deemed to have knowledge of
     any Event of Default unless the Guarantee Trustee shall have received
     written notice from the Guarantor or a Holder of the Capital Securities
     (except in the case of a payment default), or a Responsible Officer of the
     Guarantee Trustee charged with the administration of this Guarantee shall
     have obtained actual knowledge, thereof.

                                  ARTICLE III
                               GUARANTEE TRUSTEE

SECTION 3.1  Guarantee Trustee; Eligibility
             ------------------------------

          (a)  There shall at all times be a Guarantee Trustee which shall:

                 (i)   not be an Affiliate of the Guarantor; and

                 (ii)  be a corporation organized and doing business under the
          laws of the United States of America or any State or Territory thereof
          or of the District of Columbia, or Person authorized under such laws
          to exercise corporate trust powers, having a combined capital and
          surplus of at least 50 million U.S. dollars ($50,000,000), and subject
          to supervision or examination by Federal, State, Territorial or
          District of Columbia authority.  If such corporation publishes reports
          of condition at least annually, pursuant to law or to the requirements
          of the supervising or examining authority referred to above, then, for
          the purposes of this Section 3.1(a) (ii), the combined capital and
          surplus of such corporation shall

                                       8
<PAGE>

          be deemed to be its combined capital and surplus as set forth in its
          most recent report of condition so published.

          (b)  If at any time the Guarantee Trustee shall cease to be eligible
     to so act under Section 3.1(a), the Guarantee Trustee shall immediately
     resign in the manner and with the effect set out in Section 3.2(c).

          (c)  If the Guarantee Trustee has or shall acquire any "conflicting
     interest" within the meaning of Section 310(b) of the Trust Indenture Act,
     the Guarantee Trustee shall either eliminate such interest or resign to the
     extent and in the manner provided by, and subject to this Guarantee.

SECTION 3.2  Appointment, Removal and Resignation of Guarantee Trustee
             ---------------------------------------------------------

          (a)  Subject to Section 3.2(b), the Guarantee Trustee may be appointed
     or removed without cause at any time by the Guarantor except during an
     Event of Default.

          (b)  The Guarantee Trustee shall not be removed in accordance with
     Section 3.2(a) until a Successor Guarantee Trustee has been appointed and
     has accepted such appointment by written instrument executed by such
     Successor Guarantee Trustee and delivered to the Guarantor.

          (c)  The Guarantee Trustee appointed to office shall hold office until
     a Successor Guarantee Trustee shall have been appointed or until its
     removal or resignation.  The Guarantee Trustee may resign from office
     (without need for prior or subsequent accounting) by an instrument in
     writing executed by the Guarantee Trustee and delivered to the Guarantor,
     which resignation shall not take effect until a Successor Guarantee Trustee
     has been appointed and has accepted such appointment by an instrument in
     writing executed by such Successor Guarantee Trustee and delivered to the
     Guarantor and the resigning Guarantee Trustee.

          (d)  If no Successor Guarantee Trustee shall have been appointed and
     accepted appointment as provided in this Section 3.2 within 60 days after
     delivery of an instrument of removal or resignation, the Guarantee Trustee
     resigning or being removed may petition any court of competent jurisdiction
     for appointment of a Successor Guarantee Trustee.  Such court may
     thereupon, after prescribing such notice, if any, as it may deem proper,
     appoint a Successor Guarantee Trustee.

          (e)  No Guarantee Trustee shall be liable for the acts or omissions to
     act of any Successor Guarantee Trustee.

          (f)  Upon termination of this Guarantee or removal or resignation of
     the Guarantee Trustee pursuant to this Section 3.2, the Guarantor shall pay
     to the Guarantee Trustee all amounts owing to the Guarantee Trustee under
     Sections 7.2 and 7.3 accrued to the date of such termination, removal or
     resignation.

                                  ARTICLE IV
                                   GUARANTEE

                                       9
<PAGE>

SECTION 4.1  Guarantee
             ---------

          (a)  The Guarantor irrevocably and unconditionally agrees to pay in
full to the Holders the Guarantee Payments (without duplication of amounts
theretofore paid by the Issuer), as and when due, regardless of any defense
(except as defense of payment by the Issuer), right of set-off or counterclaim
that the Issuer may have or assert.  The Guarantor fully, knowingly and
unconditionally waives any right that the Guarantor may have to revoke this
Guarantee under Section 2815 of the California Civil Code or otherwise.  The
Guarantor's obligation to make a Guarantee Payment may be satisfied by direct
payment of the required amounts by the Guarantor to the Holders or by causing
the Issuer to pay such amounts to the Holders.

          (b)  The Guarantor hereby also agrees to assume any and all
Obligations of the Trust and in the event any such Obligation is not so assumed,
subject to the terms and conditions hereof, the Guarantor hereby irrevocably and
unconditionally guarantees to each Beneficiary the full payment, when and as
due, of any and all obligations to such Beneficiaries.  This Agreement is
intended to be for the benefit of, and to be enforceable by, all such
Beneficiaries, whether or not such Beneficiaries have received notice hereof.

SECTION 4.2  Waiver of Notice and Demand
             ---------------------------

          The Guarantor hereby waives notice of acceptance of this Guarantee and
of any liability to which it applies or may apply, presentment, demand for
payment, any right to require a proceeding first against the Issuer or any other
Person before proceeding against the Guarantor, protest, notice of nonpayment,
notice of dishonor, notice of redemption and all other notices and demands.

SECTION 4.3  Obligations Not Affected
             ------------------------

          The obligations, covenants, agreements and duties of the Guarantor
under this Guarantee shall in no way be affected or impaired by reason of the
happening from time to time of any of the following:

          (a)  the release or waiver, by operation of law or otherwise, of the
     performance or observance by the Issuer of any express or implied
     agreement, covenant, term or condition relating to the Capital Securities
     to be performed or observed by the Issuer;

          (b)  the extension of time for the payment by the Issuer of all or any
     portion of the Distributions, Redemption Price, Special Redemption Price,
     Liquidation Distribution or any other sums payable under the terms of the
     Capital Securities or the extension of time for the performance of any
     other obligation under, arising out of, or in connection with, the Capital
     Securities (other than an extension of time for payment of Distributions,
     Redemption Price, Special Redemption Price, Liquidation Distribution or
     other sum payable that results from the extension of any interest payment
     period on the Debentures or any extension of the maturity date of the
     Debentures permitted by the Indenture);

          (c)  any failure, omission, delay or lack of diligence on the part of
     the Holders to enforce, assert or exercise any right, privilege, power or
     remedy conferred on the Holders

                                       10
<PAGE>

     pursuant to the terms of the Capital Securities, or any action on the part
     of the Issuer granting indulgence or extension of any kind;

          (d)  the voluntary or involuntary liquidation, dissolution, sale of
     any collateral, receivership, insolvency, bankruptcy, assignment for the
     benefit of creditors, reorganization, arrangement, composition or
     readjustment of debt of, or other similar proceedings affecting, the Issuer
     or any of the assets of the Issuer;

          (e)  any invalidity of, or defect or deficiency in, the Capital
     Securities;

          (f)  the settlement or compromise of any obligation guaranteed hereby
     or hereby incurred; or

          (g)  any other circumstance whatsoever that might otherwise constitute
     a legal or equitable discharge or defense of a guarantor, it being the
     intent of this Section 4.3 that the obligations of the Guarantor hereunder
     shall be absolute and unconditional under any and all circumstances.

          There shall be no obligation of the Holders to give notice to, or
obtain consent of, the Guarantor with respect to the happening of any of the
foregoing.

SECTION 4.4  Rights of Holders
             -----------------

          (a)  The Holders of a Majority in liquidation amount of the Capital
     Securities have the right to direct the time, method and place of
     conducting any proceeding for any remedy available to the Guarantee Trustee
     in respect of this Guarantee or to direct the exercise of any trust or
     power conferred upon the Guarantee Trustee under this Guarantee; provided,
                                                                      --------
     however, that (subject to Section 2.1) the Guarantee Trustee shall have the
     -------
     right to decline to follow any such direction if the Guarantee Trustee
     shall determine that the actions so directed would be unjustly prejudicial
     to the Holders not taking part in such direction or if the Guarantee
     Trustee being advised by counsel determines that the action or proceeding
     so directed may not lawfully be taken or if the Guarantor Trustee in good
     faith by its board of directors or trustees, executive committees or a
     trust committee of directors or trustees and/or Responsible Officers shall
     determine that the action or proceedings so directed would involve the
     Guarantee Trustee in personal liability.

          (b)  Any Holder of Capital Securities may institute a legal proceeding
     directly against the Guarantor to enforce the Guarantee Trustee's rights
     under this Guarantee, without first instituting a legal proceeding against
     the Issuer, the Guarantee Trustee or any other Person.  The Guarantor
     waives any right or remedy to require that any such action be brought first
     against the Issuer, the Guarantee Trustee or any other Person before so
     proceeding directly against the Guarantor.

SECTION 4.5  Guarantee of Payment
             --------------------

          This Guarantee creates a guarantee of payment and not of collection.

                                       11
<PAGE>

SECTION 4.6  Subrogation
             -----------

          The Guarantor shall be subrogated to all (if any) rights of the
Holders of Capital Securities against the Issuer in respect of any amounts paid
to such Holders by the Guarantor under this Guarantee; provided, however, that
                                                       --------  -------
the Guarantor shall not (except to the extent required by mandatory provisions
of law) be entitled to enforce or exercise any right that it may acquire by way
of subrogation or any indemnity, reimbursement or other agreement, in all cases
as a result of payment under this Guarantee, if, after giving effect to any such
payment, any amounts are due and unpaid under this Guarantee.  If any amount
shall be paid to the Guarantor in violation of the preceding sentence, the
Guarantor agrees to hold such amount in trust for the Holders and to pay over
such amount to the Holders.

SECTION 4.7  Independent Obligations
             -----------------------

          The Guarantor acknowledges that its obligations hereunder are
independent of the obligations of the Issuer with respect to the Capital
Securities and that the Guarantor shall be liable as principal and as debtor
hereunder to make Guarantee Payments pursuant to the terms of this Guarantee
notwithstanding the occurrence of any event referred to in subsections (a)
through (g), inclusive, of Section 4.3 hereof.

SECTION 4.8  Enforcement by a Beneficiary
             ----------------------------

          A Beneficiary may enforce the obligations of the Guarantor contained
in Section 4.1(b) directly against the Guarantor and the Guarantor waives any
right or remedy to require that any action be brought against the Trust or any
other person or entity before preceding against the Guarantor.  The Guarantor
shall be subrogated to all rights (if any) of any Beneficiary against the Trust
in respect of any amounts paid to the Beneficiaries by the Guarantor under this
Agreement; provided, however, that the Guarantor shall not (except to the extent
           --------  -------
required by mandatory provisions of law) be entitled to enforce or exercise any
rights that it may acquire by way of subrogation or any indemnity, reimbursement
or other agreement, in all cases as a result of payment under this Agreement,
if, after giving effect to such payment, any amounts are due and unpaid under
this Agreement.


                                   ARTICLE V
                   LIMITATION OF TRANSACTIONS; SUBORDINATION

SECTION 5.1  Limitation of Transactions
             --------------------------

          So long as any Capital Securities remain outstanding, if (a) there
shall have occurred and be continuing an Event of Default or a Declaration Event
of Default or (b) the Guarantor shall have selected an Extension Period as
provided in the Declaration and such period, or any extension thereof, shall be
continuing, then the Guarantor may not (x) declare or pay any dividends or
distributions on, or redeem, purchase, acquire, or make a liquidation payment
with respect to, any of the Guarantor's capital stock or (y) make any payment of
principal of or interest or premium, if any, on or repay, repurchase or redeem
any debt securities of the Guarantor (including Other Debentures) that rank pari
passu in all respects with or junior in interest to the Debentures or (iii) make
any guarantee payments with respect to any guarantee

                                       12
<PAGE>

by the Company of the debt securities of any subsidiary of the Company
(including Other Guarantees) if such guarantee ranks pari passu with or junior
in right of payment to the Debentures (other than (i) repurchases, redemptions
or other acquisitions of shares of capital stock of the Guarantor in connection
with any employment contract, benefit plan or other similar arrangement with or
for the benefit of one or more employees, officers, directors or consultants, in
connection with a dividend reinvestment or stockholder stock purchase plan or in
connection with the issuance of capital stock of the Guarantor (or securities
convertible into or exercisable for such capital stock) as consideration in an
acquisition transaction entered into prior to occurrence of the Event of Default
or Declaration Event of Default or the applicable Extension Period, (ii) as a
result of any exchange or conversion of any class or series of the Guarantor's
capital stock (or any capital stock of a subsidiary of the Guarantor) for any
class or series of the Guarantor's capital stock or of any class or series of
the Guarantor's indebtedness for any class or series of the Guarantor's capital
stock, (iii) the purchase of fractional interests in shares of the Guarantor's
capital stock pursuant to the conversion or exchange provisions of such capital
stock or the security being converted or exchanged, (iv) any declaration of a
dividend in connection with any stockholder's rights plan, or the issuance of
rights, stock or other property under any stockholder's rights plan, or the
redemption or repurchase of rights pursuant thereto, or (v) any dividend in the
form of stock, warrants, options or other rights where the dividend stock or the
stock issuable upon exercise of such warrants, options or other rights is the
same stock as that on which the dividend is being paid or ranks pari passu with
or junior to such stock).

SECTION 5.2  Ranking
             -------

          This Guarantee will constitute an unsecured obligation of the
Guarantor and will rank subordinate and junior in right of payment to all
present and future Senior Indebtedness (as defined in the Indenture) of the
Guarantor.  By their acceptance thereof, each Holder of Capital Securities
agrees to the foregoing provisions of this Guarantee and the other terms set
forth herein, it being understood that the terms of Article XV of the Indenture
shall apply to the obligations of the Guarantor under this guarantee as if such
Article were set forth herein in full.

          The right of the Guarantor to participate in any distribution of
assets of any of its subsidiaries upon any such subsidiary's liquidation or
reorganization or otherwise is subject to the prior claims of creditors of that
subsidiary, except to the extent the Guarantor may itself be recognized as a
creditor of that subsidiary.  Accordingly, the Guarantor's obligations under
this Guarantee will be effectively subordinated to all existing and future
liabilities of the Guarantor's subsidiaries, and claimants should look only to
the assets of the Guarantor for payments thereunder.  This Guarantee does not
limit the incurrence or issuance of other secured or unsecured debt of the
Guarantor, including Senior Indebtedness of the Guarantor, under any indenture
that the Guarantor may enter into in the future or otherwise.

                                  ARTICLE VI
                                  TERMINATION

SECTION 6.1  Termination
             -----------

          This Guarantee shall terminate as to the Capital Securities (i) upon
full payment of the Redemption Price of all Capital Securities, (ii) upon the
distribution of the Debentures to

                                       13
<PAGE>

the Holders of all of the Capital Securities or (iii) upon full payment of the
amounts payable in accordance with the Declaration upon dissolution, winding up
or liquidation of the Issuer. This Guarantee will continue to be effective or
will be reinstated, as the case may be, if at any time any Holder of Capital
Securities must restore payment of any sums paid under the Capital Securities or
under this Guarantee.

                                  ARTICLE VII
                                INDEMNIFICATION

SECTION 7.1  Exculpation
             -----------

          (a)  No Indemnified Person shall be liable, responsible or accountable
     in damages or otherwise to the Guarantor or any Covered Person for any
     loss, damage or claim incurred by reason of any act or omission performed
     or omitted by such Indemnified Person in good faith in accordance with this
     Guarantee and in a manner that such Indemnified Person reasonably believed
     to be within the scope of the authority conferred on such Indemnified
     Person by this Guarantee or by law, except that an Indemnified Person shall
     be liable for any such loss, damage or claim incurred by reason of such
     Indemnified Person's negligence or willful misconduct with respect to such
     acts or omissions.

          (b)  An Indemnified Person shall be fully protected in relying in good
     faith upon the records of the Issuer or the Guarantor and upon such
     information, opinions, reports or statements presented to the Issuer or the
     Guarantor by any Person as to matters the Indemnified Person reasonably
     believes are within such other Person's professional or expert competence
     and who, if selected by such Indemnified Person, has been selected with
     reasonable care by such Indemnified Person, including information,
     opinions, reports or statements as to the value and amount of the assets,
     liabilities, profits, losses, or any other facts pertinent to the existence
     and amount of assets from which Distributions to Holders of Capital
     Securities might properly be paid.

SECTION 7.2  Indemnification
             ---------------

          (a)  The Guarantor agrees to indemnify each Indemnified Person for,
and to hold each Indemnified Person harmless against, any and all loss,
liability, damage, claim or expense incurred without negligence or willful
misconduct on the part of the Indemnified Person, arising out of or in
connection with the acceptance or administration of the trust or trusts
hereunder, including the costs and expenses (including reasonable legal fees and
expenses) of the Indemnified Person defending itself against, or investigating,
any claim or liability in connection with the exercise or performance of any of
the Indemnified Person's powers or duties hereunder. The obligation to indemnify
as set forth in this Section 7.2 shall survive the resignation or removal of the
Guarantee Trustee and the termination of this Guarantee.

          (b)  Promptly after receipt by an Indemnified Person under this
Section 7.2 of notice of the commencement of any action, such Indemnified Person
will, if a claim in respect thereof is to be made against the Guarantor under
this Section 7.2, notify the Guarantor in writing of the commencement thereof;
but the failure so to notify the Guarantor (i) will not

                                       14
<PAGE>

relieve the Guarantor from liability under paragraph (a) above unless and to the
extent that the Guarantor did not otherwise learn of such action and such
failure results in the forfeiture by the Guarantor of substantial rights and
defenses and (ii) will not, in any event, relieve the Guarantor from any
obligations to any Guarantor other than the indemnification obligation provided
in paragraph (a) above. The Guarantor shall be entitled to appoint counsel of
the Guarantor's choice at the Guarantor's expense to represent the Indemnified
Person in any action for which indemnification is sought (in which case the
Guarantor shall not thereafter be responsible for the fees and expenses of any
separate counsel retained by the Indemnified Person or Persons except as set
forth below); provided, however, that such counsel shall be satisfactory to the
              --------  -------
Indemnified Person. Notwithstanding the Guarantor's election to appoint counsel
to represent the Guarantor in an action, the Indemnified Person shall have the
right to employ separate counsel (including local counsel), and the Guarantor
shall bear the reasonable fees, costs and expenses of such separate counsel if
(i) the use of counsel chosen by the Guarantor to represent the Indemnified
Person would present such counsel with a conflict of interest, (ii) the actual
or potential defendants in, or targets of, any such action include both the
Indemnified Person and the Guarantor and the Indemnified Person shall have
reasonably concluded that there may be legal defenses available to it and/or
other Indemnified Person which are different from or additional to those
available to the Guarantor, (iii) the Guarantor shall not have employed counsel
satisfactory to the Indemnified Person to represent the Indemnified Person
within a reasonable time after notice of the institution of such action or (iv)
the Guarantor shall authorize the Indemnified Person to employ separate counsel
at the expense of the Guarantor. The Guarantor will not, without the prior
written consent of the Indemnified Persons, settle or compromise or consent to
the entry of any judgment with respect to any pending or threatened claim,
action, suit or proceeding in respect of which indemnification or contribution
may be sought hereunder (whether or not the Indemnified Persons are actual or
potential parties to such claim or action) unless such settlement, compromise or
consent includes an unconditional release of each Indemnified Person from all
liability arising out of such claim, action, suit or proceeding.

SECTION 7.3  Compensation; Reimbursement of Expenses
             ---------------------------------------

          The Guarantor agrees:

          (a)  to pay to the Guarantee Trustee from time to time such
compensation for all services rendered by it hereunder as the parties shall
agree to from time to time (which compensation shall not be limited by any
provision of law in regard to the compensation of a trustee of an express
trust); and

          (b)  except as otherwise expressly provided herein, to reimburse the
Guarantee Trustee upon request for all reasonable expenses, disbursements and
advances incurred or made by it in accordance with any provision of this
Guarantee (including the reasonable compensation and the expenses and
disbursements of its agents and counsel), except any such expense, disbursement
or advance as may be attributable to its negligence or willful misconduct.

          The provisions of this Section 7.3 shall survive the resignation or
removal of the Guarantee Trustee and the termination of this Guarantee.

                                       15
<PAGE>

                                 ARTICLE VIII
                                 MISCELLANEOUS

SECTION 8.1  Successors and Assigns
             ----------------------

          All guarantees and agreements contained in this Guarantee shall bind
the successors, assigns, receivers, trustees and representatives of the
Guarantor and shall inure to the benefit of the Holders of the Capital
Securities then outstanding.  Except in connection with any merger or
consolidation of the Guarantor with or into another entity or any sale, transfer
or lease of the Guarantor's assets to another entity, in each case, to the
extent permitted under the Indenture, the Guarantor may not assign its rights or
delegate its obligations under this Guarantee without the prior approval of the
Holders of at least a Majority in liquidation amount of the Capital Securities.

SECTION 8.2  Amendments
             ----------

          Except with respect to any changes that do not adversely affect the
rights of Holders of the Capital Securities in any material respect (in which
case no consent of Holders will be required), this Guarantee may be amended only
with the prior approval of the Holders of not less than a Majority in
liquidation amount of the Capital Securities.  The provisions of the Declaration
with respect to amendments thereof apply to the giving of such approval.

SECTION 8.3  Notices
             -------

          All notices provided for in this Guarantee shall be in writing, duly
signed by the party giving such notice, and shall be delivered, telecopied or
mailed by first class mail, as follows:

          (a)  If given to the Guarantee Trustee, at the Guarantee Trustee's
     mailing address set forth below (or such other address as the Guarantee
     Trustee may give notice of to the Holders of the Capital Securities):

                 The Bank of New York
                 101 Barclay Street, Floor 21W
                 New York, NY 10286
                 Attention: Corporate Trust Administration
                 Telecopy: 212-815-5915

          (b)  If given to the Guarantor, at the Guarantor's mailing address set
     forth below (or such other address as the Guarantor may give notice of to
     the Holders of the Capital Securities and to the Guarantee Trustee):

                 400 Emerson Street
                 3rd Floor
                 Palo Alto, CA 94301
                 Attention: Linda M. Iannone
                 Telecopy: 650-473-9419

                                       16
<PAGE>

          (c)  If given to any Holder of the Capital Securities, at the address
     set forth on the books and records of the Issuer.

          All such notices shall be deemed to have been given when received in
person, telecopied with receipt confirmed, or mailed by first class mail,
postage prepaid except that if a notice or other document is refused delivery or
cannot be delivered because of a changed address of which no notice was given,
such notice or other document shall be deemed to have been delivered on the date
of such refusal or inability to deliver.

SECTION 8.4  Benefit
             -------

          This Guarantee is solely for the benefit of the Holders of the Capital
Securities and, subject to Section 2.1(a), is not separately transferable from
the Capital Securities.

SECTION 8.5  Governing Law
             -------------

          THIS GUARANTEE SHALL BE GOVERNED BY, AND CONSTRUED AND INTERPRETED IN
ACCORDANCE WITH, THE LAW OF THE STATE OF NEW YORK, WITHOUT REGARD TO CONFLICT OF
LAWS PRINCIPLES THEREOF.

                                       17
<PAGE>

           THIS GUARANTEE is executed as of the day and year first above
written.

                                GREATER BAY BANCORP,
                                as Guarantor

                                By: /s/ Steven C. Smith
                                    --------------------------------
                                    Name:  Steven C. Smith
                                    Title: Executive Vice President,
                                           Chief Administrative Officer and
                                           Chief Financial Officer

                                THE BANK OF NEW YORK,
                                as Guarantee Trustee

                                By: /s/ Annette L. Kos
                                    --------------------------------
                                    Name:  Annette L. Kos
                                    Title: Assistant Vice President

                                       18