<PAGE>

                                                                    Exhibit 10.6


     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 5,771 SHARES OF
           COMMON STOCK, PAR VALUE $0.001 PER SHARE ("COMMON STOCK")

       For VALUE RECEIVED, Special Situations Fund III, L.P. ("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"), 5,771 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
<PAGE>

Agreement. 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

                                       2
<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

                                       4
<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.

                                       5
<PAGE>

       Section 12. Identity of Transfer Agent. The Transfer Agent for the Common
                   --------------------------
Stock is American Stock Transfer & Trust Company, 40 Wall Street, NewYork, 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
                   -------
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 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.

                                       6
<PAGE>

                                       CAMBRIDGE HEART, INC.



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

                                       7
<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         Signature:
with the name of the registered holder                 ------------------------
as written on the first page of the          ------------------------------
Warrant in every particular, without         Name (please print)
alteration or enlargement or any change
whatever, unless the Warrant has been        ------------------------------
assigned.                                    ------------------------------
                                             Address
                                             ------------------------------
                                             Federal Identification or
                                             Social Security No.

                                             Assignee:

                                             ______________________________
                                             ______________________________
                                             ______________________________
                                             ______________________________

                                       8