<PAGE>

                                                                    EXHIBIT 10.5



     THIS WARRANT HAS NOT BEEN REGISTERED UNDER THE SECURITIES ACT OF 1933, AS
AMENDED, AND MAY NOT BE SOLD, TRANSFERRED, PLEDGED OR HYPOTHECATED IN THE
ABSENCE OF A REGISTRATION STATEMENT COVERING THIS WARRANT UNDER SAID ACT OR AN
EXEMPTION FROM REGISTRATION UNDER SAID ACT.

     VOID AFTER 5:00 P.M. EASTERN TIME ON JUNE 9, 2003  ("EXPIRATION DATE").


                             CAMBRIDGE HEART, INC.

                     WARRANT TO PURCHASE 19,048 SHARES OF
           COMMON STOCK, PAR VALUE $0.001 PER SHARE ("Common Stock")

     For VALUE RECEIVED, Geoffrey H. Galley ("Warrantholder"), is entitled to
purchase, subject to the provisions of this Warrant, from Cambridge Heart, Inc.,
a Delaware corporation ("Company"), at any time not later than 5:00 P.M.,
Eastern time, on the Expiration Date, at an exercise price per share equal to
$7.22 (the exercise price in effect being herein called the "Warrant Price"),
19,048 shares ("Warrant Shares") of Common Stock. The number of Warrant Shares
purchasable upon exercise of this Warrant and the Warrant Price shall be subject
to adjustment from time to time as described herein.

     Section 1.  Registration.  The Company shall maintain books for the
                 ------------
transfer and registration of the Warrant. Upon the initial issuance of the
Warrant, the Company shall issue and register the Warrant in the name of the
Warrantholder.

     Section 2.  Transfers.  As provided herein, the Warrant may be transferred
                 ---------
only pursuant to a registration statement filed under the Securities Act of
1933, as amended ("Securities Act") or an exemption from registration
thereunder. Subject to such restrictions, the Company shall transfer the Warrant
from time to time upon the books to be maintained by the Company for that
purpose, upon surrender thereof for transfer properly endorsed or accompanied by
appropriate instructions for transfer upon any such transfer, and a new Warrant
shall be issued to the transferee and the surrendered Warrant shall be canceled
by the Company.

     Section 3.  Exercise of Warrant.  Subject to the provisions hereof, the
                 -------------------
Warrantholder may exercise the Warrant in whole or in part at any time upon
surrender of the Warrant, together with delivery of the duly executed Warrant
exercise form attached hereto (the "Exercise Agreement"), to the Company during
normal business hours on any business day at the Company's principal executive
offices (or such other office or agency of the Company as it may designate by
notice to the holder hereof). The Warrantholder will not be required to make any
cash payment upon exercise hereunder, but shall only be entitled to effect a
cashless exercise of this Warrant for that number of Warrant Shares indicated in
the Exercise Agreement.
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The Warrant Shares so purchased shall be deemed to be issued to the holder
hereof or such holder's designee, as the record owner of such shares, as of the
close of business on the date on which this Warrant shall have been surrendered
(or evidence of loss, theft or destruction thereof and security or indemnity
satisfactory to the Company) and the completed Exercise Agreement shall have
been delivered. Certificates for the Warrant Shares so purchased, representing
the aggregate number of shares specified in the Exercise Agreement, shall be
delivered to the holder hereof within a reasonable time, not exceeding ten (10)
business days, after this Warrant shall have been so exercised. The certificates
so delivered shall be in such denominations as may be requested by the holder
hereof and shall be registered in the name of such holder or such other name as
shall be designated by such holder. If this Warrant shall have been exercised
only in part, then, unless this Warrant has expired, the Company shall, at its
expense, at the time of delivery of such certificates, deliver to the holder a
new Warrant representing the number of shares with respect to which this Warrant
shall not then have been exercised.

     To effect the cashless exercise, the Warrantholder shall include in the
Exercise Agreement a calculation of the number of shares of Common Stock to be
issued determined by multiplying the number of Warrant Shares to which it would
otherwise be entitled by a fraction, the numerator of which shall be the
difference between the Market Price per share of the Common Stock on the date of
exercise and the Warrant Price, and the denominator of which shall be such
Market Price per share of the Common Stock. For this purpose, the "Market Price"
of the Common Stock shall be the closing price of the Common Stock as reported
by the Nasdaq National Market on the trading day first preceding the date in
question.

     Each exercise hereof shall constitute the representation and warranty of
the Warrantholder to the Company that the representations and warranties
contained in Article 5 of the Purchase Agreement (as defined below) are true and
correct in all material respects as of the time of such exercise.

     Section 4.  Compliance with the Securities Act of 1933.  Neither this
                 ------------------------------------------
Warrant nor the Common Stock issued upon exercise hereof nor any other security
issued or issuable upon exercise of this Warrant may be offered or sold except
as provided in this agreement and in conformity with the Securities Act, and
then only against receipt of an agreement of such person to whom such offer of
sale is made to comply with the provisions of this Section 4 with respect to any
resale or other disposition of such security. The Company may cause the legend
set forth on the first page of this Warrant to be set forth on each Warrant or
similar legend on any security issued or issuable upon exercise of this Warrant,
unless counsel for the Company is of the opinion as to any such security that
such legend is unnecessary.

     Section 5.  Payment of Taxes.  The Company will pay any documentary stamp
                 ----------------
taxes attributable to the initial issuance of Warrant Shares issuable upon the
exercise of the Warrant; provided, however, that the Company shall not be
required to pay any tax or taxes which may be payable in respect of any transfer
involved in the issue or delivery of any

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<PAGE>

certificates for Warrant Shares in a name other than that of the registered
holder of the Warrant in respect of which such shares are issued, and in such
case, the Company shall not be required to issue or deliver any certificate for
Warrant Shares or any Warrant until the person requesting the same has paid to
the Company the amount of such tax or has established to the Company's
satisfaction that such tax has been paid. The holder shall be responsible for
income taxes due under federal or state law, if any such tax is due.

     Section 6.  Mutilated or Missing Warrants.  In case the Warrant shall be
                 -----------------------------
mutilated, lost, stolen, or destroyed, the Company shall issue in exchange and
substitution of and upon cancellation of the mutilated Warrant, or in lieu of
and substitution for the Warrant lost, stolen or destroyed, a new Warrant of
like tenor and for the purchase of a like number of Warrant Shares, but only
upon receipt of evidence reasonably satisfactory to the Company of such loss,
theft or destruction of the Warrant, and with respect to a lost, stolen or
destroyed Warrant, reasonable indemnity or bond, if requested by the Company.

     Section 7.  Reservation of Common Stock.  The Company hereby represents and
                 ---------------------------
warrants that there have been reserved, and the Company shall at all applicable
times keep reserved until issued (if necessary) as contemplated by this Section
7, out of the authorized and unissued Common Stock, sufficient shares to provide
for the exercise of the rights of purchase represented by the Warrant. The
Company further represents and warrants that the transfer agent for the Common
Stock ("Transfer Agent"), and every subsequent transfer agent for the Common
Stock or other shares of the Company's capital stock issuable upon the exercise
of any of the right of purchase or conversion aforesaid, shall be irrevocably
authorized and directed at all times to issue such number of authorized and
unissued shares of Common Stock as shall be issuable upon the proper exercise
hereof. The Company agrees that all Warrant Shares issued upon exercise of the
Warrant shall be, at the time of delivery of the certificates for such Warrant
Shares, duly authorized, validly issued, fully paid and non-assessable shares of
Common Stock of the Company. The Company will keep a conformed copy of this
Warrant on file with the Transfer Agent and with every subsequent transfer agent
for the Common Stock or other shares of the Company's capital stock issuable
upon the exercise of the rights of purchase represented by the Warrant. The
Company will supply from time to time the Transfer Agent with duly executed
stock certificates required to honor the outstanding Warrant.

     Section 8.  Adjustments.  Subject and pursuant to the provisions of this
                 -----------
Section 8, the Warrant Price and number of Warrant Shares subject to this
Warrant shall be subject to adjustment from time to time as set forth
hereinafter.

           (a)  If the Company shall at any time or from time to time while the
     Warrant is outstanding, pay a dividend or make a distribution on its Common
     Stock in shares of Common Stock, subdivide its outstanding shares of Common
     Stock into a greater number of shares or combine its outstanding shares
     into a smaller number of shares or issue by reclassification of its
     outstanding shares of Common Stock any shares of its capital stock
     (including any such reclassification in connection with a consolidation or
     merger in which the

                                       3
<PAGE>

     Company is the continuing corporation), then the number of Warrant Shares
     purchasable upon exercise of the Warrant and the Warrant Price in effect
     immediately prior to the date upon which such change shall become
     effective, shall be adjusted by the Company so that the Warrantholder
     thereafter exercising the Warrant shall be entitled to receive the number
     of shares of Common Stock or other capital stock which the Warrantholder
     would have received if the Warrant had been exercised immediately prior to
     such event. Such adjustment shall be made successively whenever any event
     listed above shall occur.

           (b)  If any capital reorganization, reclassification of the capital
     stock of the Company, consolidation or merger of the Company with another
     corporation in which the Company is not the survivor, or sale, transfer or
     other disposition of all or substantially all of the Company's properties
     to another corporation shall be effected, then, as a condition of such
     reorganization, reclassification, consolidation, merger, sale, transfer or
     other disposition, lawful and adequate provision shall be made whereby each
     Warrantholder shall thereafter have the right to purchase and receive upon
     the basis and upon the terms and conditions herein specified and in lieu of
     the Warrant Shares immediately theretofore issuable upon exercise of the
     Warrant, such shares of stock, securities or properties as would have been
     issuable or payable with respect to or in exchange for a number of Warrant
     Shares equal to the number of Warrant Shares immediately theretofore
     issuable upon exercise of the Warrant, had such reorganization,
     reclassification, consolidation, merger, sale, transfer or other
     disposition not taken place, and in any such case appropriate provision
     shall be made with respect to the rights and interests of each
     Warrantholder to the end that the provisions hereof (including, without
     limitations, provision for adjustment of the Warrant Price) shall
     thereafter be applicable, as nearly equivalent as may be practicable in
     relation to any shares of stock, securities or properties thereafter
     deliverable upon the exercise thereof. The Company shall not effect any
     such consolidation, merger, sale, transfer or other disposition unless
     prior to or simultaneously with the consummation thereof the successor
     corporation (if other than the Company) resulting from such consolidation
     or merger, or the corporation purchasing or otherwise acquiring such assets
     or other appropriate corporation or entity shall assume the obligation to
     deliver to the holder of the Warrant such shares of stock, securities or
     assets as, in accordance with the foregoing provisions, such holder may be
     entitled to purchase and the other obligations under this Warrant. The
     provisions of this paragraph (b) shall similarly apply to successive
     reorganizations, reclassifications, consolidations, mergers, sales,
     transfers or other dispositions.

           (c)  In case the Company shall fix a payment date for the making of a
     distribution to all holders of Common Stock (including any such
     distribution made in connection with a consolidation or merger in which the
     Company is the continuing corporation) of evidences of indebtedness or
     assets (other than cash dividends or cash distributions payable out of
     consolidated earnings or earned surplus or dividends or distributions
     referred to in Section 8(a)), or subscription rights or warrants, the
     Warrant Price to be in effect after such payment date shall be determined
     by multiplying the Warrant Price in effect immediately prior to such
     payment date by a fraction, the numerator of which shall be the total
     number of shares of Common Stock outstanding multiplied by the Market Price

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<PAGE>

     per share of Common Stock (as determined pursuant to Section 3), less the
     fair market value (as determined by the Company's Board of Directors in
     good faith) of said assets or evidences of indebtedness so distributed, or
     of such subscription rights or warrants, and the denominator of which shall
     be the total number of shares of Common Stock outstanding multiplied by
     such Market Price per share of Common Stock. Such adjustment shall be made
     successively whenever such a payment date is fixed.

           (d)  If pursuant to the Purchase Agreement by and between the Company
     and the Investor named therein dated June 8, 1999 (the "Purchase
     Agreement") there is an adjustment to the Purchase Price under Section
     7.1(a), then the Warrant Price shall be reduced to a price equal to 110% of
     such per share adjusted Purchase Price. Such adjustments shall be made
     successively whenever required.

           (e)  An adjustment shall become effective immediately after the
     payment date in the case of each dividend or distribution and immediately
     after the effective date of each other event which requires an adjustment.

           (f)  In the event that, as a result of an adjustment made pursuant to
     Section 8(a), the holder of the Warrant shall become entitled to receive
     any shares of capital stock of the Company other than shares of Common
     Stock, the number of such other shares so receivable upon exercise of the
     Warrant shall be subject thereafter to adjustment from time to time in a
     manner and on terms as nearly equivalent as practicable to the provisions
     with respect to the Warrant Shares contained in this Warrant.

     Section 9.  Fractional Interest.  The Company shall not be required to
                 -------------------
issue fractions of Warrant Shares upon the exercise of the Warrant. If any
fraction of a Warrant Share would, except for the provisions of this Section, be
issuable upon the exercise of the Warrant (or specified portions thereof), the
fractional share shall be disregarded and the number of shares to be issued upon
exercise shall be the number of whole shares only.

     Section 10.  Benefits.  Nothing in this Warrant shall be construed to give
                  --------
any person, firm or corporation (other than the Company and the Warrantholder)
any legal or equitable right, remedy or claim, it being agreed that this Warrant
shall be for the sole and exclusive benefit of the Company and the
Warrantholder.

     Section 11.  Notices to Warrantholder.  Upon the happening of any event
                  ------------------------
requiring an adjustment of the Warrant Price, the Company shall forthwith give
written notice thereof to the Warrantholder at the address appearing in the
records of the Company, stating the adjusted Warrant Price and the adjusted
number of Warrant Shares resulting from such event and setting forth in
reasonable detail the method of calculation and the facts upon which such
calculation is based. The certificate of the Company's independent certified
public accountants shall be conclusive evidence of the correctness of any
computation made, absent manifest error. Failure to give such notice to the
Warrantholder or any defect therein shall not affect the legality or validity of
the subject adjustment.

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<PAGE>

     Section 12.  Identity of Transfer Agent.  The Transfer Agent for the Common
                  --------------------------
Stock is American Stock Transfer & Trust Company, 40 Wall Street, New York,
New York 10005. Forthwith upon the appointment of any subsequent transfer agent
for the Common Stock or other shares of the Company's capital stock issuable
upon the exercise of the rights of purchase represented by the Warrant, the
Company will mail to the Warrantholder a statement setting forth the name and
address of such transfer agent.

     Section 13.  Notices.  Any notice pursuant hereto to be given or made by
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the Warrantholder to or on the Company shall be sufficiently given or made if
sent by certified mail, return receipt requested, postage prepaid, addressed as
follows:

                Cambridge Heart, Inc.
                One Oak Park Drive
                Bedford, MA  01730
                Attention:  Jeffrey M. Arnold

or such other address as the Company may specify in writing by notice to the
Warrantholder complying as to delivery with the terms of this Section 13.

     Any notice pursuant hereto to be given or made by the Company to or on the
Warrantholder shall be sufficiently given or made if personally delivered or if
sent by an internationally recognized courier services by overnight or two-day
service, to the address set forth on the books of the Company or, as to each of
the Company and the Warrantholder, at such other address as shall be designated
by such party by written notice to the other party complying as to delivery with
the terms of this Section 13. All such notices, requests, demands, directions
and other communications shall, when sent by courier be effective three (3) days
after delivery to such courier as provided and addressed as aforesaid.

     Section 14.  Registration Rights.  The initial holder of this Warrant is
                  -------------------
entitled to the benefit of certain registration rights in respect of the Warrant
Shares as provided in the Registration Rights Agreement dated as of as of
June 8, 1999.

     Section 15.   Successors.  All the covenants and provisions hereof by or
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for the benefit of the Warrantholder shall bind and inure to the benefit of its
respective successors and assigns hereunder.

     Section 16.  Governing Law.  This Warrant shall be deemed to be a contract
                  -------------
made under the laws of the State of Delaware and for all purposes shall be
construed in accordance with the laws of said State.

     IN WITNESS WHEREOF, the parties hereto have caused this Warrant to be duly
executed, as of the day and year first above written.

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<PAGE>

                                        CAMBRIDGE HEART, INC.



                                        By: /s/ Jeffrey Arnold
                                            ------------------------------
                                            Name:  Jeffrey Arnold
                                            Title: Chief Executive Officer

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<PAGE>

                             CAMBRIDGE HEART, INC.
                             WARRANT EXERCISE FORM


CAMBRIDGE HEART, INC.

---------------------

---------------------

     This undersigned hereby irrevocably elects to exercise the right of
purchase represented by the within Warrant ("Warrant") for, and to purchase
thereunder by [_] cash; or [_] cashless exercise of the within Warrant by
surrender of the Warrant,                 shares of Common Stock ("Warrant
                          ---------------
Shares") provided for therein, and requests that certificates for the Warrant
Shares be issued as follows:

                --------------------------------
                Name

                --------------------------------
                Address
                --------------------------------

                --------------------------------

                --------------------------------
                Federal Tax Identification No.
                or Social Security No.

and, if the number of Warrant Shares shall not be all the Warrant Shares
purchasable upon exercise of the Warrant, that a new Warrant for the balance of
the Warrant Shares purchasable upon exercise of the Warrant be registered in the
name of the undersigned Warrantholder or the undersigned's Assignee as below
indicated and delivered to the address stated below.


Dated:                   ,
       ------------------  ----

Note:  The signature must correspond with
                                               Signature:
the name of the registered holder as written             -----------------------
on the first page of the Warrant in every
particular, without alteration or enlargement  -------------------------------
or any change whatever, unless the Warrant     Name (please print)
has been assigned.                             -------------------------------

                                               -------------------------------

                                               -------------------------------
                                               Address
                                               ______________________________
                                               Federal Identification or
                                               Social Security No.

                                               Assignee:

                                               -------------------------------
                                               -------------------------------
                                               -------------------------------
                                               -------------------------------

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