<PAGE>   1

                                                                  EXHIBIT 10.29



                      GARDEN STATE CABLEVISION L.P. WARRANT

THIS WARRANT AND THE SECURITIES ISSUABLE UPON EXERCISE OF THIS WARRANT HAVE NOT
BEEN REGISTERED UNDER THE UNITED STATES SECURITIES ACT OF 1933, AS AMENDED (THE
"ACT"), OR UNDER THE SECURITIES LAWS OF ANY STATES OF THE UNITED STATES. THESE
SECURITIES ARE SUBJECT TO RESTRICTIONS ON TRANSFERABILITY AND RESALE AND MAY NOT
BE TRANSFERRED OR RESOLD EXCEPT PURSUANT TO REGISTRATION UNDER THE ACT OR
PURSUANT TO AN EXEMPTION THEREFROM, AND EXCEPT AS PERMITTED UNDER APPLICABLE
STATE SECURITIES LAWS. INVESTORS SHOULD BE AWARE THAT THEY MAY BE REQUIRED TO
BEAR THE FINANCIAL RISKS OF THIS INVESTMENT FOR AN INDEFINITE PERIOD OF TIME.
THE ISSUER OF THESE SECURITIES MAY REQUIRE AN OPINION OF COUNSEL IN FORM AND
SUBSTANCE SATISFACTORY TO THE ISSUER TO THE EFFECT THAT ANY PROPOSED TRANSFER OR
RESALE IS IN COMPLIANCE WITH THE ACT AND ANY APPLICABLE STATE SECURITIES LAWS.

Issue Date: June 27, 1998


                    WARRANT TO PURCHASE SERIES A COMMON STOCK
                                       OF
                               AT HOME CORPORATION

        THIS CERTIFIES THAT in consideration of the continued rapid deployment
of the @Home Service of At Home Corporation, a Delaware corporation (the
"COMPANY"), Garden State Cablevision L.P. (the "INITIAL REGISTERED HOLDER"), or
its permitted registered assigns (including the Initial Registered Holder, the
"REGISTERED HOLDER"), is entitled, subject to the terms and conditions of this
Warrant, to purchase from the Company at any time (i) on or after the
Commencement Date and (ii) on or prior to the Expiration Date, 592,528 shares of
the Company's Series A Common Stock (subject to adjustment as set forth in the
Company's Certificate of Incorporation), at an exercise price equal to $10.50
per share of Series A Common Stock (such price, as it may be adjusted pursuant
to the provisions of Section 5 below, referred to as the "EXERCISE PRICE"), upon
surrender of this Warrant at the principal office of the Company, together with
a duly executed subscription form in the form attached hereto as Exhibit 1 and
simultaneous payment of the full exercise price for the shares of Warrant Stock
so purchased. The Exercise Price and the number and kind of shares of Warrant
Stock purchasable under this Warrant are subject to adjustment as provided
herein.

Notwithstanding anything to the contrary contained in this Warrant, this Warrant
and all rights to purchase Warrant Stock hereunder shall terminate on the
Expiration Date.



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        1. CERTAIN DEFINITIONS.

               1.1 Definitions Incorporated by Reference. The following terms
shall have the meaning ascribed to them in the @Home Network Distribution
Agreement, dated as of June 27, 1998, between the Company and the Initial
Registered Holder:


<TABLE>
<CAPTION>
               Term                                       Cross Reference
               ----                                       ---------------
<S>                                                      <C> 
               @Home Service                              Section 1(d)
               Living Unit                                Section 1(q)
               Master Roll-Out Plan                       Section 2(a)(i)
               MDU                                        Section 1(u)
               Service Area                               Section 1(dd)
               Service Area Plan                          Section 2 (b)(i)
               Two-Way Data-Ready Cable System            Section 1(gg)
</TABLE>


               1.2 Additional Definitions. The following additional definitions
shall apply for purposes of this Warrant:

               "ACT" means the Securities Act of 1933, as amended.

               "COMMENCEMENT DATE" means March 31, 1999.

               "COMMERCIALLY DEPLOYED" means a Eligible Living Unit that is: (i)
connected to a Two-Way Data-Ready Cable System; and (ii) able to subscribe to
the @Home Service, if desired. For the purposes of this definition, a Living
Unit will be considered "connected" to a Two-Way Data Ready Cable System if the
Eligible Living Unit is located within 150 feet of the applicable Two-Way Data
Ready Cable System. An Eligible Living Unit in a MDU that meets the two criteria
set forth above shall be counted as one Commercially Deployed Eligible Living
Unit.

               "DETERMINATION DATE" means March 31 of each of 1999, 2000, 2001,
2002 and 2003.

               "DISTRIBUTION AGREEMENT" means the @Home Network Distribution
Agreement, dated as of June 27, 1998, between the Company and the Initial
Registered Holder.

               "EXPIRATION DATE" means 5:00 p.m. Pacific Time on June 26, 2004.

               "ELIGIBLE LIVING UNITS" means the Living Units in the Service
Area indicated on the Master Roll-Out Plan that were used for calculating the
number of Warrant Shares.

               "ISSUE DATE" means the date of this Warrant.







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               "MASTER ROLL-OUT PLAN" means the Master Roll-Out Plan attached as
an exhibit to the Distribution Agreement.

               "PERSON" means any individual, corporation, limited liability
company, partnership, joint venture, association, joint stock company, trust,
unincorporated organization, government or agency or political subdivision
thereof, or other entity, whether acting in an individual, fiduciary or other
capacity.

               "SEC" means the U.S. Securities and Exchange Commission.

               "SERIES A COMMON STOCK" means the Company's Series A Common
Stock, par value $0.01 per share, and stock of any other series or class into
which the same may be changed.

               "WARRANT" means this Warrant and any warrant(s) delivered in
substitution or exchange therefor, as provided herein.

               "WARRANT PURCHASE AGREEMENT" means the Agreement dated of even
date herewith between the Company and the Initial Registered Holder relating to
the purchase of this Warrant.

               "WARRANT STOCK" means shares of Series A Common Stock issued upon
exercise of this Warrant.

        2. EXERCISE.

               2.1 Exercisability of Warrant. This Warrant is not immediately
exercisable. On each Determination Date, this Warrant shall become exercisable
as to a number of shares of Warrant Stock equal to (x) the number of Living
Units which, on such date, are (A) subject to a Service Area Plan and (B)
Commercially Deployed, multiplied by (y) two. On or following a Determination
Date, the Registered Holder shall provide the Company with a certificate,
executed by either the Chief Executive Officer or Chief Financial Officer of the
Registered Holder, setting forth in detail its calculation of such number of
residences, which determination shall be conclusive unless disputed by the
Company. In the event that the Company shall dispute such determination, the
Company shall, within three business days of its receipt of the Registered
Holder's certificate, give written notice to the Registered Holder, setting
forth in detail the basis of its dispute. In the event of such a dispute, no
shares of Warrant Stock shall become exercisable in connection with such
Determination Date until both the Company and the Registered Holder shall have
resolved such dispute to their mutual satisfaction. For the avoidance of doubt,
shares of Warrant Stock exercisable prior to such Determination Date shall
continue to be exercisable regardless of any such dispute. In no event shall the
number of shares of Warrant Stock exercisable pursuant to this Warrant exceed
592,528.

               2.2 Surrender. Subject to compliance with all applicable
securities laws and the provisions of Section 2.1, this Warrant may be exercised
in whole or in part by surrendering this Warrant at the principal office of the
Company at 425 Broadway, Redwood City, California



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<PAGE>   4


94063, with the subscription form attached hereto as Exhibit 1 duly executed by
the Registered Holder, accompanied by payment as set forth in Section 2.3 below.

               2.3 Payment of Exercise Price. Payment shall be made at any time
with respect to shares of Warrant Stock being purchased hereunder (x) by the
payment to the Company, by cash, check and/or wire transfer, of an amount equal
to the then-applicable Exercise Price per share multiplied by the number of
shares of Warrant Stock then being purchased, or, at the option of the
Registered Holder, (y) by surrendering to the Company for cancellation the right
to receive upon exercise hereof a number of shares of Series A Common Stock
equal to the value (as determined below) of the shares of Warrant Stock with
respect to which this Warrant is being exercised, in which case the number of
shares to be issued to the Registered Holder upon such exercise shall be
computed using the following formula:

               X =    Y(A-B)
                      ------
                        A

Where:         X =    the number of shares of Series A Common Stock to be issued
                      to the Registered Holder.

               Y =    the number of shares of Series A Common Stock with
                      respect to which this Warrant is being exercised and with
                      respect to which the right to receive shares is being
                      canceled.

               A =    the fair market value of one share of Series A Common
                      Stock.

               B =    the Exercise Price per share of Series A Common Stock (as
                      it may be adjusted pursuant to the provisions of Section
                      5);

provided, that in the case of a cashless exercise pursuant to clause (y), the
Registered Holder shall only be entitled to surrender for cancellation the right
to receive shares which may then be issued upon exercise of this Warrant.

As used herein, the "fair market value of one share of Series A Common Stock"
shall mean the average, for the five trading days (or such fewer number of days
as the Company's Series A Common Stock may have been publicly traded) ending
with the trading day which is two trading days prior to the date of such
surrender, of:

                    (a) the closing prices of the Company's Series A Common
Stock sold on the securities exchange(s) on which the Series A Common Stock may
at the time be listed, or

                    (b) if there have been no sales on such exchange(s) on any
such trading day, the average of the highest bid and lowest asked prices on such
exchange(s) at the end of such day, or




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                    (c) if on any such trading day the Series A Common Stock is
not so listed, the average of the representative bid and asked prices quoted on
the Nasdaq National Market ("NASDAQ") as of 4:00 p.m., New York City time, on
such day, or

                    (d) if on any such trading day the Series A Common Stock is
not quoted on Nasdaq, the average of the highest bid and lowest asked price on
such day in the domestic over-the-counter market as reported by the National
Quotation Bureau, Incorporated, or any similar successor organization.

               2.4 Date of Exercise; Fractional Shares. Except as otherwise
provided in Section 2.3, this Warrant shall be deemed to have been exercised
immediately prior to the close of business on the date of its surrender for
exercise as provided in Sections 2.2 and 2.3, and the person entitled to receive
the shares of Warrant Stock issuable upon such exercise shall be treated for all
purposes as the holder of record of such shares as of the close of business on
such date. As soon as practicable on or after such date, the Company shall issue
and deliver to the person or persons entitled to receive the same a certificate
or certificates for the number of whole shares of Warrant Stock issuable upon
such exercise, together with cash in lieu of any fraction of a share equal to
such fraction of the current fair market value of one whole share of Warrant
Stock as of the date of exercise, as determined in good faith by the Company's
Board of Directors. No fractional shares may be issued upon any exercise of this
Warrant, and any fractions shall be rounded down to the nearest whole number of
shares.

               2.5 Partial Exercise. Upon a partial exercise of this Warrant,
this Warrant shall be surrendered by the Registered Holder and replaced with a
new Warrant of like tenor in the name of the Registered Holder providing for the
right to purchase the number of shares of Warrant Stock as to which this Warrant
has not then been exercised.

               2.6 Taxes. The issuance of certificates for shares of Warrant
Stock upon the exercise of this Warrant will be made without charge by the
Company to the Registered Holder for any issue tax (other than applicable income
tax).

        3. REPRESENTATIONS AND COVENANTS OF THE COMPANY.

               3.1 Fully Paid Shares; Reservation. The Company hereby represents
and warrants to the Registered Holder that all shares of Warrant Stock which may
be issued upon exercise of this Warrant shall have been duly and validly
reserved for issuance and, upon issuance, be duly authorized, validly issued,
fully paid and nonassessable, and free of any liens, claims, charges, security
interests, pledges or encumbrances of any kind, except for restrictions on
transfer provided for in this Warrant and under applicable federal and state
securities laws. If at any time the number of authorized but unissued shares of
the Company's Warrant Stock shall not be sufficient to effect the exercise of
this Warrant, the Company will take such corporate action as may, in the opinion
of its counsel, be necessary to increase its authorized but unissued shares of
Warrant Stock to such number of shares of Warrant Stock as shall be sufficient
for such purpose.




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<PAGE>   6


               3.2 Notices. The Company agrees that it will notify the
Registered Holder at least ten (10) business days in advance of the proposed
consummation of any pending consolidation or merger of the Company into any
other corporation or the sale of all or substantially all of the Company's
assets to another corporation, unless after the closing of any such transaction
the stockholders of the Company immediately prior to such transaction own in
excess of fifty percent (50%) of the voting power of the surviving corporation
or its parent corporation. Such notice shall include a description of all
material terms and conditions of such transaction and the per share value of the
consideration to be paid in connection therewith, and other information given by
the Company to the holders of its Series A Common Stock in connection with their
approval thereof.

               3.3 No Impairment. The Company will not, by amendment of its
Certificate of Incorporation or Bylaws, or through reorganization,
consolidation, merger, dissolution, issue or sale of securities, sale of assets
or any other voluntary action, willfully avoid or seek to avoid the observance
or performance of any of the terms of this Warrant, but will at all times in
good faith assist in the carrying out of all such terms and in the taking of all
such action as may be necessary or appropriate in order to protect the rights of
the Registered Holder under this Warrant against wrongful impairment. Without
limiting the generality of the foregoing, the Company: (i) will not set nor
increase the par value of any shares of stock issuable upon exercise of this
Warrant above the amount payable therefor upon such exercise, and (ii) will take
all such action as may be necessary or appropriate in order that the Company may
validly and legally issue fully paid and non-assessable shares of Warrant Stock
upon the exercise of this Warrant.

        4. TRANSFER RESTRICTIONS.

               4.1 Limitations on Transfer.

                   (i) Any portion of this Warrant that is then exercisable
pursuant to Section 2.1 above may be assigned, conveyed or transferred by the
Registered Holder to a third party without the Company's consent provided that
the third party agrees in writing to comply with all of the terms and conditions
of this Warrant and the Company is informed in writing of such assignment,
conveyance or transfer.

                   (ii) Unless otherwise agreed by @Home in writing, which
consent will not be unreasonably withheld, any portion of this Warrant that is
not exercisable may not be assigned, conveyed or transferred by the Registered
Holder to a third party except when the Eligible Homes underlying that portion
of the Warrant are transferred to that third party in accordance with the
Distribution Agreement, such third party agrees in writing to comply with all of
the terms and conditions of this Warrant, and the Company is informed in writing
of such assignment, conveyance or transfer.

               4.2 Mechanics and Effects of Transfer. Any assignment, conveyance
or transfer of the Warrant, the Warrant Stock or the rights hereunder shall be
made on the books of the Company maintained for such purpose at the principal
office of the Company upon surrender of this Warrant or the Warrant Stock and a
properly completed assignment in the form of Exhibit



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<PAGE>   7


2 hereto. All transferees under this Section 4 will be bound by the provisions
of this Section 4. Notwithstanding the foregoing, this Warrant and the rights
hereunder may not be assigned, conveyed or transferred unless such assignment,
conveyance or transfer also complies with all applicable securities laws and the
provisions of Section 8.2 hereof.

               4.3 Legends; Notations. The certificates evidencing the Warrant
Stock shall be endorsed with the legends set forth below:

                    (a) a conspicuously noted legend in substantially the
following form:

               "THE SECURITIES REPRESENTED BY THIS CERTIFICATE HAVE NOT BEEN
REGISTERED UNDER THE SECURITIES ACT OF 1933, AS AMENDED (THE "ACT"), OR ANY
STATE SECURITIES LAWS AND MAY NOT BE SOLD, TRANSFERRED, PLEDGED, HYPOTHECATED OR
OTHERWISE DISPOSED OF UNLESS AND UNTIL REGISTERED UNDER THE ACT AND ANY
APPLICABLE STATE SECURITIES LAWS OR UNLESS, IN THE OPINION OF COUNSEL REASONABLY
SATISFACTORY TO THE COMPANY, SUCH SALE, TRANSFER, PLEDGE, HYPOTHECATION OR OTHER
DISPOSITION IS EXEMPT FROM REGISTRATION OR IS OTHERWISE IN COMPLIANCE WITH THE
ACT AND SUCH LAWS."; and

                    (b) any legend required by any applicable state securities
law.

               The Company shall make a notation on its stock books regarding
the restrictions on transfer of the Warrants and Warrant Stock provided by
applicable securities and other laws and this Warrant, and will transfer
securities on the books of the Company only to the extent not inconsistent
therewith. Without limiting the foregoing, the Company shall refuse to register
any transfer of the Warrants or Warrant Stock not made in accordance with or
pursuant to an applicable exemption from registration under the Act and
applicable state securities laws.

        5. ADJUSTMENT OF EXERCISE PRICE, NUMBER AND KIND OF SHARES. The number
and kind of shares of Warrant Stock issuable upon exercise of this Warrant (or
any shares of stock or other securities or property at the time receivable or
issuable upon exercise of this Warrant) and the Exercise Price therefor, are
subject to adjustment upon the occurrence of the following events:

               5.1 Adjustment for Stock Splits, Stock Dividends,
Recapitalizations, etc. The Exercise Price of this Warrant and the number of
shares of Series A Common Stock issuable upon exercise of this Warrant shall
each be proportionally adjusted to reflect any stock dividend, stock split,
reverse stock split, recapitalization and the like affecting the number of
outstanding shares of Series A Common Stock that occurs after the Issue Date.

               5.2 Adjustment for Reorganization, Consolidation, Merger. In case
of any reorganization of the Company (or of any other corporation, the stock or
other securities of which are at the time receivable on the exercise of this
Warrant), after the Issue Date, or in case, after such date, the Company (or any
such corporation) shall consolidate with or merge into another corporation or
convey all or substantially all of its assets to another corporation or other



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<PAGE>   8


entity, then, and in each such case, the Registered Holder of this Warrant, upon
any permitted exercise of this Warrant (as provided in Section 2), at any time
after the consummation of such reorganization, consolidation, merger, or
conveyance, shall be entitled to receive, in lieu of the stock or other
securities and property receivable upon the exercise of this Warrant prior to
such consummation, the stock or other securities or property to which such
Registered Holder would have been entitled upon the consummation of such
reorganization, consolidation, merger or conveyance if such Registered Holder
had exercised this Warrant immediately prior thereto, all subject to further
adjustment as provided in this Section 5, and the successor or purchasing
corporation or other entity in such reorganization, consolidation, merger or
conveyance (if other than the Company) shall duly execute and deliver to the
Registered Holder a supplement hereto acknowledging such corporation's or
entity's obligations under this Warrant; and in each such case, the terms of
this Warrant (including the exercisability, transfer and adjustment provisions
of this Warrant) shall be applicable to the shares of stock or other securities
or property receivable upon the exercise of this Warrant after the consummation
of such reorganization, consolidation, merger or conveyance.

        6. CERTIFICATE AS TO ADJUSTMENTS. In each case of any adjustment in
either the Exercise Price or in the number of shares of Warrant Stock, or other
stock, securities or property receivable upon the exercise of this Warrant, the
Chief Financial Officer of the Company shall promptly thereafter compute such
adjustment in accordance with the terms of this Warrant and prepare a
certificate setting forth such adjustment and showing in detail the facts upon
which such adjustment is based, including a statement of the adjusted Exercise
Price. The Company will cause copies of such certificate to be mailed (by first
class mail, postage prepaid) to the Registered Holder.

        7. LOSS OR MUTILATION. Upon receipt by the Company of evidence
reasonably satisfactory to it of the ownership, and the loss, theft, destruction
or mutilation, of this Warrant, and of indemnity reasonably satisfactory to it,
and (in the case of mutilation) upon surrender and cancellation of this Warrant,
the Company will execute and deliver in lieu thereof a new Warrant of like
tenor.

        8. REPRESENTATIONS AND WARRANTIES OF THE REGISTERED HOLDER.

               8.1 Restrictions under Securities Laws. The Registered Holder
understands that neither the offer and sale of this Warrant nor the offer and
sale of shares of Warrant Stock that may be purchased upon exercise thereof have
been registered under the Act, or any state securities laws. As a condition to
the issuance of this Warrant and to its exercise the Registered Holder hereby
represents and warrants to the Company that:

                      (a) The Warrant and the underlying shares of Warrant Stock
(collectively, the "SECURITIES") are being and/or will be acquired by the
Registered Holder in a transaction exempt from registration under Section 4(2)
of the Act and/or Regulation D promulgated under the Act, for its own account,
for investment purposes only, and not with a view to the sale or other
distribution thereof within the meaning of the Act and the Registered



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<PAGE>   9

Holder has no present intention of selling or otherwise disposing of all or any
portion of the Securities except as permitted by the Warrant Purchase Agreement
and this Warrant.

                      (b) The Registered Holder is capable of evaluating the
merits and risks of any investment in the Securities, is financially capable of
bearing a total loss of this investment and either: (i) has a preexisting
personal or business relationship with the Company or its principals; (ii) by
reason of the Registered Holder's business or financial experience, has the
capacity to protect his or its own interests in connection with this investment;
or (iii) if the Registered Holder is the Initial Registered Holder, is an
"accredited investor" within the meaning of Regulation D promulgated under the
Act, as amended.

                      (c) The Registered Holder has had access to all
information regarding the Company, its present and prospective business, assets,
liabilities and financial condition that the Registered Holder considers
important to making the decision to acquire the Securities and has had ample
opportunity to ask questions of and receive answers from the Company's
representatives concerning an investment in the Securities and to obtain any and
all documents requested in order to supplement or verify any of the information
supplied.

                      (d) The Registered Holder understands that the Securities
shall be deemed restricted securities under the Act and may not be resold unless
they are registered under the Act and any applicable State securities law, or in
the opinion of counsel in form and substance satisfactory to the Company, an
exemption from such registration is available.

                      (e) The Registered Holder is aware of Rule 144 promulgated
under the Act, which rule provides, in substance, that: (i) after one year from
the date restricted securities have been purchased and fully paid for, a holder
may transfer restricted securities provided certain conditions are met (e.g.,
certain public information is available about the Company), and specific
limitations on the amount of shares which can be sold within certain periods and
the manner in which such shares must be sold are complied with; and (ii) after
two years from the date the securities have been purchased and fully paid for,
holders who are not "affiliates" of the Company may sell restricted securities
without satisfying such conditions.

                      (f) The Registered Holder further understands that if the
requirements of Rule 144 are not met, registration under the Act or compliance
with some other registration exemption will be required for any disposition of
the Securities; and that, although Rule 144 is not exclusive, the SEC has
expressed its opinion that persons proposing to sell restricted securities other
than in a registered offering or other than pursuant to Rule 144 will have a
substantial burden of proof in establishing that an exemption from registration
is available for such offers or sales and such persons and the brokers who
participate in the transactions do so at their own risk. The Registered Holder
understands that the Company is under no obligation to register the Securities
or take any other actions under the Act or any state securities laws.

               8.2 Compliance with Securities Laws. The Registered Holder of
this Warrant, by acceptance hereof, agrees that, absent an effective
registration statement filed with the SEC under the Act covering the disposition
or sale of any Securities, such Registered Holder will not



                                      -9-
<PAGE>   10


sell or transfer any or all of such Securities unless such sale or transfer is
pursuant to an available exemption from registration under the Act and
applicable state securities laws. As a condition to any such sale or transfer,
the Registered Holder shall first provide the Company with an opinion of counsel
satisfactory to the Company to the effect that such sale or transfer is or will
be exempt from the registration and prospectus delivery requirements of the Act
and applicable state securities laws. Such Registered Holder consents to the
Company making a notation on its records, or giving instructions to any transfer
agent of such Securities, in order to implement the foregoing restrictions on
transfer. The shares issued upon exercise of this Warrant shall bear legends
referring to the restrictions on transfer set forth in this Section 8. As a
condition to the transfer of this Warrant or transfer of the shares issuable on
exercise hereof, any permitted transferee must execute and deliver to the
Company representations and warranties similar to these set forth in this
Section 8 and applicable to a transferee of securities in an exempt transaction
under the Act and must agree in writing to accept and be bound by all the terms
and conditions of this Warrant.

        9. NO RIGHTS OR LIABILITIES AS STOCKHOLDER. This Warrant does not by
itself entitle the Registered Holder to any voting rights or other rights as a
stockholder of the Company. In the absence of affirmative action by the
Registered Holder to purchase Warrant Stock by exercise of this Warrant, no
provisions of this Warrant, and no enumeration herein of the rights or
privileges of the Registered Holder shall cause such Registered Holder to be a
stockholder of the Company for any purpose.

        10. REGULATORY COMPLIANCE. If the Registered Holder of this Warrant or
the Warrant Stock would be subject to the Hart-Scott-Rodino Antitrust
Improvements Act of 1976 and the rules and regulations thereunder (collectively,
the "ANTITRUST LAW"), then prior to such exercise or conversion and following
such Registered Holder's notice to the Company of its intention to exercise or
convert, the Company and such Registered Holder shall promptly use commercially
reasonable efforts to comply with any applicable requirements under the
Antitrust Law relating to filing and furnishing of information to the Federal
Trade Commission and the Antitrust Division of the Department of Justice. Each
of the Company and such Registered Holder shall bear and pay any costs or
expenses that it incurs in compliance with this requirement.

        11. AMENDMENT; WAIVER. Any term of this Warrant may be amended and the
observance of any term of this Warrant may be waived (either generally or in a
particular instance and either retroactively or prospectively), only with the
written consent of the Company and the Registered Holder. Any amendment or
waiver effected in accordance with this Section 11 shall be binding upon the
Registered Holder, any future Registered Holder and the Company.

        12. NOTICES. All notices and other communications from the Company to
the Registered Holder shall be deemed given when personally delivered, mailed by
first-class registered or certified mail, postage prepaid, delivered by
recognized overnight courier service, or transmitted by facsimile (with
confirmation by first class mail), to the address furnished to the Company in
writing by the Registered Holder who shall have furnished an address and/or
facsimile number to the Company in writing.




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<PAGE>   11

        13. HEADINGS. The headings in this Warrant are for purposes of
convenience in reference only, and shall not be deemed to constitute a part
hereof.

        14. LAW GOVERNING; DISPUTE RESOLUTION. This Warrant shall be construed
and enforced in accordance with, and governed by, the internal laws of the State
of California, excluding that body of law applicable to conflicts of laws. Any
claim, controversy or dispute between the parties relating to this Warrant (a
"Dispute") will be resolved in accordance with the following procedure:

               (i) The Chief Executive Officers of each of the parties involved
in the Dispute will first discuss the Dispute and use their good faith efforts
to reach a fair and equitable resolution.

               (ii) In the event the Chief Executive Officers of the parties
involved in the Dispute are unable to resolve the Dispute, the Board of
Directors of @Home will review the Dispute and use its good faith efforts to
reach a fair and equitable resolution.

               (iii) In the event the Board of Directors of @Home are unable to
resolve the dispute to the satisfaction of any of the parties involved in the
Dispute, the dissatisfied party may submit the Dispute to a confidential
arbitration proceeding. Such arbitration proceeding will be held in San
Francisco, California will be subject to the rules of the American Arbitration
Association ("AAA"), and will be conducted by a single arbitrator chosen by the
AAA. The parties agree that the decision of the arbitrator will be final and
binding upon the parties.

        15. TERMS BINDING. By acceptance of this Warrant, the Registered Holder
of this Warrant (and each subsequent assignee, transferee or Registered Holder
of this Warrant) accepts and agrees to be bound by all the terms and conditions
of this Warrant.

        16. COUNTERPARTS. This Warrant may be executed in one or more
counterparts, each of which shall be an original and all of which together shall
constitute one and the same instrument.


        IN WITNESS WHEREOF, the Company and the Registered Holder have executed
this Warrant as of the Issue Date.


THE COMPANY:                                ACKNOWLEDGED AND ACCEPTED
                                            BY INITIAL REGISTERED HOLDER:

AT HOME CORPORATION                         GARDEN STATE CABLEVISION L.P.

By: /s/ Kenneth A. Goldman                  By: /s/ Brian W. Earnshaw
   ------------------------------              -----------------------------

Name:                                       Name: Brian W. Earnshaw
     ----------------------------                ---------------------------


Title:                                      Title: Vice President - Finance 
      ---------------------------                 --------------------------

                                                   and Administration
                                                  --------------------------

                                                   October 8, 1998
                                                  --------------------------

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