EXHIBIT 4.1(b)


          THIS FIRST SUPPLEMENTAL WARRANT AGREEMENT, dated as of July 17, 2003
(the "Supplement"), between The Bear Stearns Companies Inc., a Delaware
corporation (the "Company"), JPMorgan Chase Bank, as Warrant Agent (the "Warrant
Agent"), and Bear, Stearns & Co. Inc., as Calculation Agent (the "Calculation
Agent"). Capitalized terms used but not defined herein shall have the meanings
set forth in the Warrant Agreement (as defined below).

                             RECITALS OF THE COMPANY

          WHEREAS, the Company and the Warrant Agent have previously entered
into a Warrant Agreement dated as of July 9, 2003 (the "Warrant Agreement");

          WHEREAS, the Company desires to enter into this Supplement in order to
issue Warrants pursuant to the terms of the Warrant Agreement, including Section
1.03 thereof; and

          NOW, THEREFORE, it is mutually covenanted and agreed as follows:

                                   ARTICLE I
                                 WARRANT DETAILS

          Section 1.01. The title of the Warrants to be issued under this
Supplement is "Accelerated Market Participation Securities Linked to the S&P 500
Index" (the "AMPS"). The aggregate principal amount of the AMPS which may be
delivered under this Supplement is limited to $5,000,000. The AMPS will be
issued only as registered securities at a purchase price of $5.00 per Warrant on
July 23, 2003. The AMPS will be issued in global form, and the Depository for
such AMPS in global form shall be The Depository Trust Company ("DTC").
Beneficial interests in such AMPS will be held through DTC. The interests in
such AMPS in global form may be exchanged in whole for AMPS which represent the
individual AMPS at the option of (i) the Company or (ii) the Depository if the
Depository is unwilling or unable to continue as depository and a successor
depository is not appointed by the Company within 90 calendar days. In any such
instance, an owner of a beneficial interest in the AMPS in global form will be
entitled to physical delivery in definitive form of the AMPS equal in principal
amount to such beneficial interest and to have such AMPS registered in its name.

          The AMPS will be automatically exercised on October 21, 2004, the
expiration date (the "Expiration Date"). There shall not be any periodic
payments on the AMPS or any other payments on the AMPS until the Expiration
Date.

          On the Expiration Date, a holder of an AMPS will receive a payment
equal to the Cash Settlement Value (as defined below). The amount a
Warrantholder will receive upon exercise of the AMPS will be based upon the
percentage change in the value of the S&P 500 Composite Price Index (the
"Index"). The Cash Settlement Value to which a Warrantholder will be entitled
depends entirely on the relation of the Final Index Level to the Initial Index
Level (each as defined below).
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          The "Cash Settlement Amount" with respect to the AMPS shall be
calculated as follows:

          (i) If the Final Index Level is greater than or equal to the Initial
Index Level, the Cash Settlement Value per AMPS will equal:


            /              /  Final Index Level-Initial Index Level  \  \
     $5 +  |  ($5)(3)  x  |   -------------------------------------   |  |
            \              \           Initial Index Level           /  /


          provided, however, the Cash Settlement Value cannot exceed $5.75 per
AMPS.

          (ii) If the Final Index Level is less than the Initial Index Level,
the Cash Settlement Value per AMPS will equal:


                               /   Final Index Level   \
                       $5  x  |   -------------------   |
                               \  Initial Index Level  /


          For purposes of this formula:

          The "Initial Index Level" equals 981.73, the closing value of the
Index on July 17, 2003, the date the AMPS were priced for initial sale to the
public.

          The "Final Index Level" will be determined by the Calculation Agent
and will equal the closing value of the Index on October 18, 2004, the
"valuation date".

          If the Calculation Agent determines that there is a Market Disruption
Event (as defined below) on the valuation date, the valuation date will be the
first succeeding Index Business Day on which there is no Market Disruption
Event, unless there is a Market Disruption Event on each of the five Index
Business Days following the original date that, but for the Market Disruption
Event, would have been the valuation date. In that case, the fifth Index
Business Day will be deemed to be the valuation date, notwithstanding the Market
Disruption Event, and the Calculation Agent will determine the level of the
Index on that fifth Index Business Day in accordance with the formula for and
method of calculating the Index in effect prior to the Market Disruption Event
using the exchange traded price of each security in the Index (or, if trading in
any such security has been suspended or materially limited, the Calculation
Agent's good faith estimate of the exchange traded price that would have
prevailed but for such suspension or limitation) as of that fifth Index Business
Day.

          An "Index Business Day" means a day on which The Nasdaq Stock Market,
the New York Stock Exchange (the "NYSE") and The American Stock Exchange are
open for trading and the Index or any successor index is calculated and
published.

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<PAGE>

          A "Market Disruption Event" means either of the following events, as
determined by the Calculation Agent, in its sole discretion:

(a)  the suspension of or material limitation on trading for more than two hours
     of trading, or during the one-half hour period preceding the close of
     trading on the applicable exchange in 20% or more of the stocks which then
     comprise the Index (without taking into account any extended or after-hours
     trading session); or

(b)  the suspension of or material limitation on trading, in each case, for more
     than two hours of trading, or during the one-half hour period preceding the
     close of trading, on the applicable exchange, whether by reason of
     movements in price otherwise exceeding levels permitted by the relevant
     exchange or otherwise, in option contracts or futures contracts related to
     the Index, or any successor index, which are traded on any major U.S.
     exchange.

          For the purpose of the above definition:

o    a limitation on the hours in a trading day and/or number of days of trading
     will not constitute a Market Disruption Event if it results from an
     announced change in the regular business hours of the relevant exchange,
     and

o    for the purpose of clause (a) above, any limitations on trading during
     significant market fluctuations under NYSE Rule 80A, or any applicable rule
     or regulation enacted or promulgated by the NYSE or any other self
     regulatory organization or the Securities Exchange Commission of similar
     scope as determined by the calculation agent, will be considered
     "material".

          If Standard & Poor's discontinues publication of the Index, or if it
or another entity publishes a successor or substitute index that the Calculation
Agent determines, in its sole discretion, to be comparable to the Index, then
the ending value as of any succeeding valuation date will be determined by
reference to the value of that index, a "successor index." Upon any selection by
the Calculation Agent of a successor index, the Calculation Agent will cause
notice to be furnished to the Company and the Warrant Agent, who will provide
notice of the selection of the successor index to the registered holders of the
AMPS. If Standard & Poor's discontinues publication of the Index, and a
successor index is not selected by the Calculation Agent, or is no longer
published on any valuation date, the periodic index level to be substituted for
the Index for that valuation date will be a value computed by the Calculation
Agent for that valuation date in accordance with the procedures last used to
calculate the Index prior to any such discontinuance. If Standard & Poor's
discontinues publication of the Index prior to the determination of the Cash
Settlement Value, and the Calculation Agent determines that no successor index
is available at that time, then on each Index Business Day until the earlier to
occur of (a) the determination of the Cash Settlement Value or (b) a
determination by the Calculation Agent that a successor index is available, the
Calculation Agent will determine the value that is to be used in computing the
Cash Settlement Value as described in the preceding paragraph, as if such day
were a valuation date. The Calculation Agent will cause notice of each such
value to be published not less often than once each month in The Wall Street
Journal (or another newspaper of general circulation), and arrange for
information with respect to those values to be made available by telephone. If a
successor index is selected, or the Calculation

                                      -3-
<PAGE>

Agent calculates a value as a substitute for the Index as described above, the
successor index or value will be substituted for the Index for all purposes,
including for purposes of determining whether an Index Business Day or Market
Disruption Event has occurred.

          Section 1.02. Each AMPS will be automatically exercised in whole but
not in part on the Expiration Date, without any requirement of notice of
exercise to the Warrant Agent. The Final Index Level and Cash Settlement Value
of such Warrants shall be determined on the Expiration Date, or, if the
Expiration Date is not a Business Day, then on the Business Day following the
Expiration Date. As used herein, "Business Day" means any day other than a
Saturday or a Sunday or a day on which banking institutions in New York City are
authorized or required by law, regulation or executive order to close.

          (a) On the Expiration Date or, if the Expiration Date is not a
Business Day, then on the Business Day next following the Expiration Date, the
Calculation Agent shall (i) obtain the Final Index Level (which Final Index
Level shall be the Final Index Level on such day) from the Calculation Agent,
(ii) determine the Cash Settlement Value (in the manner provided above) of the
Warrants to be automatically exercised, (iii) advise the Company by 5:00 P.M.,
New York City time, on the Expiration Date or, if the Expiration Date is not a
Business Day, then on the Business Day next following the Expiration Date, of
the Cash Settlement Value with respect to such Warrants and (iv) advise the
Company of such other matters relating to the automatically exercised Warrants
as the Company shall reasonably request. Provided that the Company has made
adequate funds available to the Warrant Agent in a timely manner which shall, in
no event, be later than 3:00 P.M., New York City time, on the third Business Day
following the date on which the Final Index Level is obtained, the Warrant Agent
will make payment available (i) in the case of Global Warrant Certificates, in
same day funds or other acceptable means to the Depository, against receipt of
the Global Warrant Certificate, and (ii) in the case of Definitive Warrant
Certificates, in the form of a cashier's check or an official bank check to the
appropriate Registered Holder, or (in the case of payments of at least $250,000)
by wire transfer to a U.S. dollar account maintained by such Registered Holder
(at the Registered Holder's written election), against receipt by the Warrant
Agent at the Warrant Agent's Office from such Registered Holder of its Warrant
Certificates, in either case after 3:00 P.M., New York City time, but prior to
the close of business, on such third Business Day following the date on which
the Final Index Level is obtained, such check to be in the amount of the
aggregate Cash Settlement Value in respect of Warrants that have been
automatically exercised. Warrant Certificates delivered to the Warrant Agent,
and the Global Warrant Certificate, shall thereafter be promptly cancelled by
the Warrant Agent.

          Section 1.03. The Company hereby appoints Bear, Stearns & Co. Inc. as
Calculation Agent of the Company in respect of the AMPS upon the terms and
subject to the conditions set forth in the Warrant Agreement; and Bear, Stearns
& Co. Inc. hereby accepts such appointment. The Calculation Agent shall have the
powers and authority granted to and conferred upon it in the Warrant Agreement
and such further powers and authority to act on behalf of the Company as the
Company may hereafter grant to or confer upon it with its consent.

                                      -4-
<PAGE>

                                   ARTICLE II

                                  MISCELLANEOUS

          Section 2.01. This Supplement shall be effective on the date hereof.

          Section 2.02. The recitals contained herein shall be taken as the
statements of the Company, and the Warrant Agent assumes no responsibility for
their correctness. The Warrant Agent makes no representations as to the validity
or sufficiency of this Supplement.

          Section 2.03. This Supplement shall be governed by and construed in
accordance with the laws of the jurisdiction which govern the Warrant Agreement
and its construction.

          Section 2.04. All agreements of the Company in this Supplement shall
bind its successors. All agreements of the Warrant Agent in this Supplement
shall bind its successors.

          Section 2.05. This Supplement may be executed in any number of
counterparts each of which shall be an original, but such counterparts shall
together constitute but one and the same instrument.


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<PAGE>

                  IN WITNESS WHEREOF, the parties hereto have caused this
Supplement to be duly executed and their respective seals to be affixed hereunto
and duly attested all as of the day and year first above written.

                                The Bear Stearns Companies Inc.



                                By /s/ Samuel L. Molinaro Jr.
                                   --------------------------------------------
                                   Name:  Samuel L. Molinaro Jr.
                                   Title: Executive Vice President and
                                          Chief Financial Officer


                                JPMorgan Chase Bank, as Warrant Agent



                                By  /s/ Natalie Pesce
                                   --------------------------------------------
                                   Name:  Natalie Pesce
                                   Title: Trust Officer


                                Bear, Stearns & Co. Inc., as Calculation Agent



                                By  /s/ Harry Engelman
                                   --------------------------------------------
                                   Name:  Harry Engelman
                                   Title: Senior Managing Director


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