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                                                                   EXHIBIT 10.61

THESE SECURITIES HAVE NOT BEEN REGISTERED WITH THE UNITED STATES SECURITIES AND
EXCHANGE COMMISSION OR THE SECURITIES COMMISSION OF ANY STATE PURSUANT TO AN
EXEMPTION FROM REGISTRATION UNDER REGULATION D PROMULGATED UNDER THE SECURITIES
ACT OF 1933, AS AMENDED (THE "ACT"). THIS WARRANT SHALL NOT CONSTITUTE AN OFFER
TO SELL NOR A SOLICITATION OF AN OFFER TO BUY THE SECURITIES IN ANY JURISDICTION
IN WHICH SUCH OFFER OR SOLICITATION WOULD BE UNLAWFUL. THE SECURITIES ARE
"RESTRICTED" AND MAY NOT BE RESOLD OR TRANSFERRED EXCEPT AS PERMITTED UNDER THE
ACT PURSUANT TO REGISTRATION OR EXEMPTION THEREFROM.

THIS WARRANT DOES NOT REQUIRE PHYSICAL SURRENDER OF THE WARRANT IN THE EVENT
THAT IT IS EXERCISED. AS A RESULT, FOLLOWING ANY EXERCISE OF ANY PORTION OF THIS
WARRANT, THIS WARRANT MAY BE EXERCISED FOR A NUMBER OF WARRANT SHARES WHICH IS
LESS THAN THE NUMBER SET FORTH BELOW. ACCORDINGLY, ANY POTENTIAL ASSIGNEE MAY
WISH TO, BUT IS NOT OBLIGATED TO, CONFIRM THE NUMBER OF WARRANT SHARES THEN
REPRESENTED HEREBY WITH THE COMPANY, IN ADDITION TO THE ASSIGNOR.

Issue Date:  June 5, 2000

                          COMMON STOCK PURCHASE WARRANT

     To Purchase Shares of $0.01 Par Value Common Stock ("COMMON STOCK") of

                            APPLIEDTHEORY CORPORATION

         THIS CERTIFIES that, for value received, HALIFAX FUND, L.P. (the
"INVESTOR") is entitled, upon the terms and subject to the conditions
hereinafter set forth, at any time on or after the Issue Date and on or prior to
8:00 p.m. New York City Time on the five (5) year anniversary of the Issue Date
(the "TERMINATION DATE"), but not thereafter, to subscribe for and purchase from
APPLIEDTHEORY CORPORATION, a Delaware corporation (the "COMPANY"), a number of
shares of Common Stock (the "WARRANT SHARES") of the Company equal to 1,083,333
at an Exercise Price per share equal to $6.00 (as adjusted from time to time
pursuant to the terms hereof, the "EXERCISE PRICE"). The Exercise Price and the
number of shares for which the Warrant is exercisable shall be subject to
adjustment as provided herein. This Warrant is being issued in connection with
the Amended and Restated Purchase Agreement dated as of June 5, 2000 (the
"PURCHASE AGREEMENT") entered into between the Company, the Investor, and
certain other parties. Capitalized terms used herein and not otherwise defined
shall have the meaning ascribed thereto in the Purchase Agreement.

1.       Title of Warrant. Prior to the expiration hereof and subject to
         compliance with applicable laws, this Warrant and all rights hereunder
         are transferable, in whole or in part, at the office or agency of the
         Company by the Holder hereof in person or by duly authorized attorney,
         upon surrender of this Warrant together with (a) the Assignment Form
         annexed hereto properly endorsed, and (b) any other documentation
         reasonably necessary to satisfy


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         the Company that such transfer is in compliance with all applicable
         securities laws. The term "HOLDER" shall refer to the Investor or any
         subsequent transferee of this Warrant.

2.       Authorization of Shares. The Company covenants that all shares of
         Common Stock which may be issued upon the exercise of rights
         represented by this Warrant will, upon exercise of the rights
         represented by this Warrant and payment of the Exercise Price as set
         forth herein will be duly authorized, validly issued, fully paid and
         nonassessable and free from all taxes, liens and charges in respect of
         the issue thereof (other than taxes in respect of any transfer
         occurring contemporaneously with such issue or otherwise specified
         herein).

3.       Exercise of Warrant.

(a)      (i)      The Holder may exercise this Warrant, in whole or in part, at
                  any time and from time to time, by delivering to the offices
                  of the Company or any transfer agent for the Common Stock (x)
                  a Notice of Exercise in the form annexed hereto specifying the
                  number of Warrant Shares with respect to which this Warrant is
                  being exercised, (y) payment to the Company of the Exercise
                  Price therefor and (z) this Warrant, subject to the Holder
                  indicating that it is effecting delivery of this Warrant
                  pursuant to 3(a)(v).

         (ii)     In the event that the Warrant is not exercised in full, the
                  number of Warrant Shares shall be reduced by the number of
                  such Warrant Shares for which this Warrant is exercised and/or
                  surrendered, and the Company, at its expense, shall within
                  three (3) Trading Days (as defined below) issue and deliver to
                  the Holder a new Warrant of like tenor in the name of the
                  Holder or as the Holder (upon payment by Holder of any
                  applicable transfer taxes) may request, reflecting such
                  adjusted Warrant Shares, subject to the provisions of Section
                  3(a)(v) below.

                  The term "TRADING DAY" means (x) if the Common Stock is not
                  listed on the New York or American Stock Exchange but sale
                  prices of the Common Stock are reported on Nasdaq National
                  Market or another automated quotation system, or the Nasdaq
                  Bulletin Board, a day on which trading is reported on the
                  principal automated quotation system or market on which sales
                  of the Common Stock are reported, (y) if the Common Stock is
                  listed on the New York Stock Exchange or the American Stock
                  Exchange, a day on which there is trading on such stock
                  exchange, or (z) if the foregoing provisions are inapplicable,
                  a day on which quotations are reported by National Quotation
                  Bureau Incorporated.


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         (iii)    Subject to the provisions of Section 3(a)(iv) below, the
                  Company shall use its best efforts to deliver the certificates
                  for shares of Common Stock purchased hereunder to the Holder
                  hereof within three (3) Trading Days after the date on which
                  this Warrant shall have been exercised as aforesaid. The
                  Holder may withdraw its Notice of Exercise at any time if the
                  Company fails to deliver within 10 calendar days the relevant
                  certificates to the Holder as provided in this Agreement.

         (iv)     In lieu of delivering physical certificates representing the
                  Warrant Shares issuable upon exercise of this Warrant,
                  provided the Company's transfer agent is participating in the
                  Depository Trust Company ("DTC") Fast Automated Securities
                  Transfer ("FAST") program, upon request of the Holder, the
                  Company shall use its best efforts to cause its transfer agent
                  to electronically transmit the Warrant Shares issuable upon
                  exercise to the Holder, by crediting the account of the
                  Holder's prime broker with DTC through its Deposit Withdrawal
                  Agent Commission ("DWAC") system. The time periods for
                  delivery described above shall apply to the electronic
                  transmittals through the DWAC system. The Company agrees to
                  coordinate with DTC to accomplish this objective.

         (v)      Notwithstanding anything to the contrary set forth herein,
                  upon exercise of any portion of this Warrant in accordance
                  with the terms hereof, the Holder shall not be required to
                  physically surrender this Warrant to the Company unless such
                  Holder is purchasing the full amount of Warrant Shares
                  represented by this Warrant. The Holder and the Company shall
                  maintain records showing the number of Warrant Shares so
                  purchased hereunder and the dates of such purchases or shall
                  use such other method, reasonably satisfactory to the Holder
                  and the Company, so as not to require physical surrender of
                  this Warrant upon each such exercise. In the event that the
                  Holder has exercised this Warrant in part pursuant to the
                  foregoing provisions, and subsequently wishes to transfer this
                  Warrant, the Holder may not subsequently transfer this Warrant
                  unless the Holder first physically surrenders this Warrant to
                  the Company, whereupon the Company will forthwith cancel the
                  Warrant and issue and deliver upon the order of the Holder a
                  new Warrant of like tenor, registered as the Holder may
                  request, representing in the aggregate the remaining number of
                  Warrant Shares which may be purchased upon exercise of this
                  Warrant and reflecting the use of the book entry provisions
                  set forth herein. The Holder and any assignee, by acceptance
                  of this Warrant or a new Warrant, acknowledge and agree that,
                  by reason of the provisions of this paragraph, following
                  exercise of any portion of this Warrant, the number of Warrant
                  Shares which may be purchased upon exercise of this Warrant
                  may be less than the number of Warrant Shares set forth on the
                  face hereof.

(b)      Cashless Exercise. Notwithstanding the foregoing provision regarding
         payment of the Exercise Price in cash, the Holder may elect to receive
         a reduced number of Warrant Shares in lieu of tendering the Exercise
         Price in cash. In such case, the number of Warrant Shares to be issued
         to the Holder shall be computed using the following formula:


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

<TABLE>
<S>             <C>
         where: X = the number of Warrant Shares to be issued to the Holder;

                Y = the number of Warrant Shares to be exercised under this
                    Warrant Certificate;

                A = the Market Value (defined below) of one share of Common Stock;
                    and

                B = the Exercise Price.
</TABLE>

         As used herein, "MARKET VALUE" refers to the closing price of the
         Common Stock (as reported by Bloomberg, L.P.) on the day before the
         Notice of Exercise and this Warrant are duly surrendered to the Company
         for a full or partial exercise hereof. Notwithstanding the foregoing
         definition, if the Common Stock is not listed on a national securities
         exchange or quoted in the Nasdaq System at the time said Notice of
         Exercise is submitted to the Company in the foregoing manner, the
         Market Value of the Common Stock shall be as reasonably determined in
         good faith by the Board of Directors of the Company and such Holder,
         unless the Company shall become subject to a merger, acquisition, or
         other consolidation pursuant to which the Company is not the surviving
         entity, in which case the Market Value of the Common Stock shall be
         deemed to be the value received by the Company's common shareholders
         pursuant to the Company's acquisition (subject to Section 12 below).

4.       No Fractional Shares or Scrip. No fractional shares or scrip
         representing fractional shares shall be issued upon the exercise of
         this Warrant. In lieu of issuance of a fractional share upon any
         exercise hereunder, the Company will either round up to nearest whole
         number of shares or pay the cash value of that fractional share
         calculated on the basis of the Fair Market Value (as defined below).

5.       Charges, Taxes and Expenses. Issuance of certificates for shares of
         Common Stock upon the exercise of this Warrant shall be made without
         charge to the Holder hereof for any issue or transfer tax or other
         incidental expense in respect of the issuance of such certificate, all
         of which taxes and expenses shall be paid by the Company, and such
         certificates shall be issued in the name of the Holder of this Warrant
         or in such name or names as may be directed by the Holder of this
         Warrant; provided, however, that in the event certificates for shares
         of Common Stock are to be issued in a name other than the name of the
         Holder of this Warrant, this Warrant when surrendered for exercise
         shall be accompanied by the Assignment Form attached hereto duly
         executed by the Holder hereof; and provided further, 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 issuance of any Warrant
         certificates or any certificates for the Warrant Shares other than the
         issuance of a Warrant Certificate to the Holder in connection with the
         Holder's surrender of a Warrant Certificate upon the exercise of all or
         less than all of the Warrants evidenced thereby.

6.       Closing of Books. The Company will at no time close its shareholder
         books or records in any manner which interferes with the timely
         exercise of this Warrant.


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7.       No Rights as Shareholder until Exercise. Subject to Section 12 of this
         Warrant and the provisions of any other written agreement between the
         Company and the Investor, the Investor shall not be entitled to vote or
         receive dividends or be deemed the holder of Warrant Shares or any
         other securities of the Company that may at any time be issuable on the
         exercise hereof for any purpose, nor shall anything contained herein be
         construed to confer upon the Investor, as such, any of the rights of a
         stockholder of the Company or any right to vote for the election of
         directors or upon any matter submitted to stockholders at any meeting
         thereof, or to give or withhold consent to any corporate action
         (whether upon any recapitalization, issuance of stock, reclassification
         of stock, change of par value, or change of stock to no par value,
         consolidation, merger, conveyance or otherwise) or to receive notice of
         meetings, or to receive dividends or subscription rights or otherwise
         until the Warrant shall have been exercised as provided herein.
         However, at the time of the exercise of this Warrant pursuant to
         Section 3 hereof, the Warrant Shares so purchased hereunder shall be
         deemed to be issued to such Holder as the record owner of such shares
         as of the close of business on the date on which this Warrant shall
         have been exercised.

8.       Assignment and Transfer of Warrant. This Warrant may be assigned by the
         surrender of this Warrant and the Assignment Form annexed hereto duly
         executed at the office of the Company (or such other office or agency
         of the Company or its transfer agent as the Company may designate by
         notice in writing to the registered Holder hereof at the address of
         such Holder appearing on the books of the Company); provided, however,
         that this Warrant may not be resold or otherwise transferred except (i)
         in a transaction registered under the Act, or (ii) in a transaction
         pursuant to an exemption, if available, from registration under the Act
         and whereby, if reasonably requested by the Company, an opinion of
         counsel reasonably satisfactory to the Company is obtained by the
         Holder of this Warrant to the effect that the transaction is so exempt.
         If this Warrant is duly assigned in accordance with the terms hereof,
         then the Company agrees, upon the request of the assignee, to amend or
         supplement promptly any effective registration statement covering the
         Warrant Shares so that the direct assignee of the original holder is
         added as a selling stockholder thereunder.

9.       Loss, Theft, Destruction or Mutilation of Warrant; Exchange. The
         Company represents warrants and covenants that (a) upon receipt by the
         Company of evidence reasonably satisfactory to it of the loss, theft,
         destruction or mutilation of any Warrant or stock certificate
         representing the Warrant Shares, and in case of loss, theft or
         destruction, of indemnity reasonably satisfactory to it, and (b) upon
         surrender and cancellation of such Warrant or stock certificate, if
         mutilated, the Company will make and deliver a new Warrant or stock
         certificate of like tenor and dated as of such cancellation, in lieu of
         this Warrant or stock certificate, without any charge therefor. This
         Warrant is exchangeable at any time for an equal aggregate number of
         Warrants of different denominations, as requested by the Holder
         surrendering the same, or in such denominations as may be requested by
         the Holder following determination of the Exercise Price. No service
         charge will be made for such registration or transfer, exchange or
         reissuance.


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10.      Saturdays, Sundays, Holidays, etc. If the last or appointed day for the
         taking of any action or the expiration of any right required or granted
         herein shall be a Saturday, Sunday or a legal holiday, then such action
         may be taken or such right may be exercised on the next succeeding day
         not a legal holiday.

11.      Specific Enforcement. The Company and the Holder acknowledge and agree
         that irreparable damage would occur in the event that any of the
         provisions of this Warrant were not performed in accordance with their
         specific terms or were otherwise breached. It is accordingly agreed
         that the parties shall be entitled to an injunction or injunctions to
         prevent or cure breaches of the provisions of this Warrant and to
         enforce specifically the terms and provisions hereof, this being in
         addition to any other remedy to which either of them may be entitled by
         law or equity.

12.      Adjustments of Exercise Price and Number of Warrant Shares. The number
         of and kind of securities purchasable upon exercise of this Warrant and
         the Exercise Price shall be subject to adjustment from time to time as
         set forth in this Section 12.

(a)      Subdivisions, Combinations, Stock Dividends and other Issuances. If the
         Company shall, at any time while this Warrant is outstanding, (A) pay a
         stock dividend or otherwise make a distribution or distributions on any
         equity securities (including instruments or securities convertible into
         or exchangeable for such equity securities) in shares of Common Stock,
         (B) subdivide outstanding shares of Common Stock into a larger number
         of shares, or (C) combine outstanding Common Stock into a smaller
         number of shares, then the Exercise Price shall be multiplied by a
         fraction, the numerator of which shall be the number of shares of
         Common Stock outstanding before such event and the denominator of which
         shall be the number of shares of Common Stock outstanding after such
         event. Any adjustment made pursuant to this Section 12(a) shall become
         effective immediately after the record date for the determination of
         stockholders entitled to receive such dividend or distribution and
         shall become effective immediately after the effective date in the case
         of a subdivision or combination. The number of shares which may be
         purchased hereunder shall be increased proportionately to any reduction
         in Exercise Price pursuant to this paragraph 12(a), so that after such
         adjustments the aggregate Exercise Price payable hereunder for the
         increased number of shares shall be the same as the aggregate Exercise
         Price in effect just prior to such adjustments.

(b)      Other Distributions. If at any time after the date hereof the Company
         distributes to holders of its Common Stock, other than as part of its
         dissolution, liquidation or the winding up of its affairs, any shares
         of its capital stock or other securities, any evidence of indebtedness
         or any of its assets or rights or warrants to subscribe for or purchase
         any security (other than those referred to in Section 12(a) above and
         Section 12(e) below) (in each case, "DISTRIBUTED PROPERTY"), then the
         number of Warrant Shares for which this Warrant is exercisable shall be
         increased to equal: (i) the number of Warrant Shares for which this
         Warrant is exercisable immediately prior to such event, (ii) multiplied
         by a fraction, (A) the numerator of which shall be the Fair Market
         Value (as defined below) per share of Common Stock on the record date
         for the dividend or distribution, and (B) the denominator of which
         shall be the Fair Market Value price per share of Common


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         Stock on the record date for the dividend or distribution minus the
         amount allocable to one share of Common Stock of the value (as jointly
         determined in good faith by the Board of Directors of the Company and
         the Holder) of any and all such distributed property. For purposes of
         this Warrant, "FAIR MARKET VALUE" shall equal the average closing price
         of the Common Stock on the Principal Market (as defined in the Purchase
         Agreement) for the 5 Trading Days preceding the date of determination
         or, if the Common Stock is not listed or admitted to trading on any
         Principal Market, and the average price cannot be determined as
         contemplated above, the Fair Market Value of the Common Stock shall be
         as reasonably determined in good faith by the Company's Board of
         Directors and the Holder. The Exercise Price shall be reduced to equal:
         (i) the Exercise Price in effect immediately before the occurrence of
         any event (ii) multiplied by a fraction, (A) the numerator of which is
         the number of Warrant Shares for which this Warrant is exercisable
         immediately before the adjustment, and (B) the denominator of which is
         the number of Warrant Shares for which this Warrant is exercisable
         immediately after the adjustment.

(c)      Merger, etc. If at any time after the date hereof there shall be a
         merger or consolidation of the Company with or into or a transfer of
         all or substantially all of the assets of the Company to another
         entity, then the Holder shall be entitled to receive upon or after such
         transfer, merger or consolidation becoming effective, and upon payment
         of the Exercise Price then in effect, the number of shares or other
         securities or property of the Company or of the successor corporation
         resulting from such merger or consolidation, all subject to adjustment
         as provided herein, which would have been received by the Holder for
         the shares of stock subject to this Warrant had this Warrant been
         exercised just prior to such transfer, merger or consolidation becoming
         effective or to the applicable record date thereof, as the case may be.
         The Company will not merge or consolidate with or into any other
         corporation, or sell or otherwise transfer its property, assets and
         business substantially as an entirety to another corporation, unless
         the corporation resulting from such merger or consolidation (if not the
         Company), or such transferee corporation, as the case may be, shall
         expressly assume in writing the due and punctual performance and
         observance of each and every covenant and condition of this Warrant to
         be performed and observed by the Company.

(d)      Reclassification, etc. If at any time after the date hereof there shall
         be a reorganization or reclassification of the securities as to which
         purchase rights under this Warrant exist into the same or a different
         number of securities of any other class or classes, then the Holder
         shall thereafter be entitled to receive upon exercise of this Warrant,
         during the period specified herein and upon payment of the Exercise
         Price then in effect, the number of shares or other securities or
         property resulting from such reorganization or reclassification, which
         would have been received by the Holder for the shares of stock subject
         to this Warrant had this Warrant at such time been exercised, all
         subject to adjustment as provided herein.

(e)      Exercise Price Adjustment. In the event that the Company issues or
         sells any Common Stock or securities which are convertible into or
         exchangeable for its Common Stock or any convertible securities, or any
         warrants or other rights to subscribe for or to purchase or any


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         options for the purchase of its Common Stock or any such convertible
         securities at an effective price per share which is less than the
         greater of the Exercise Price then in effect or the Fair Market Value
         (as described in Section 12(b) above) of the Common Stock on the
         Trading Day next preceding such issue or sale, then in each such case,
         the Exercise Price in effect immediately prior to such issue or sale
         shall be reduced effective concurrently with such issue or sale to an
         amount determined by multiplying the Exercise Price then in effect by a
         fraction, (x) the numerator of which shall be the sum of (1) the number
         of shares of Common Stock outstanding immediately prior to such issue
         or sale, plus (2) the number of shares of Common Stock which the
         aggregate consideration received by the Company for such additional
         shares would purchase at such Fair Market Value or Exercise Price,
         whichever is greater, then in effect; and (y) the denominator of which
         shall be the number of shares of Common Stock of the Company
         outstanding immediately after such issue or sale.

         For the purposes of the foregoing adjustment, in the case of the
         issuance of any convertible securities, warrants, options or other
         rights to subscribe for or to purchase or exchange for, shares of
         Common Stock ("CONVERTIBLE SECURITIES"), the maximum number of shares
         of Common Stock issuable upon exercise, exchange or conversion of such
         Convertible Securities shall be deemed to be outstanding, provided that
         no further adjustment shall be made upon the actual issuance of Common
         Stock upon exercise, exchange or conversion of such Convertible
         Securities. With respect to Convertible Securities, the Per Share
         Selling Price shall be equal to the lowest price at which shares of
         Common Stock may be issued on conversion, exercise or exchange of the
         Convertible Securities at the time the adjustment is being calculated.
         If such lowest price may change after issuance of the Convertible
         Securities (for example, by reason of adjustments or resets resulting
         from changes in the market price for the Common Stock, or for any other
         reason) then this adjustment shall be recomputed to reflect such new
         "lowest price."

         The number of shares which may be purchased hereunder shall be
         increased proportionately to any reduction in Exercise Price pursuant
         to this paragraph 12(e), so that after such adjustments the aggregate
         Exercise Price payable hereunder for the increased number of shares
         shall be the same as the aggregate Exercise Price in effect just prior
         to such adjustments.

         In the event of any such issuance for a consideration which is less
         than such Fair Market Value and also less than the Exercise Price then
         in effect, than there shall be only one such adjustment by reason of
         such issuance, such adjustment to be that which results in the greatest
         reduction of the Exercise Price computed as aforesaid. This Section
         12(e) shall not apply to issuances or sales of securities in connection
         with strategic acquisitions of other entities by the Company which
         engage in businesses related or complementary to the Company's business
         and that are not essentially capital raising transactions on behalf of
         the Company.

(f)      (i) The terms of any reorganization, consolidation, merger, sale,
         transfer or share exchange shall include such terms so as to continue
         to give to the holder hereof the right to receive the securities or
         property set forth in this Section 12 upon any exercise


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         following any such reclassification, consolidation, merger, sale,
         transfer or share exchange.

         (ii) In the event of any adjustment in the number of Warrant Shares
         issuable hereunder upon exercise, the Exercise Price shall be inversely
         proportionately increased or decreased as the case may be, such that
         aggregate purchase price for Warrant Shares upon full exercise of this
         Warrant shall remain the same. Similarly, in the event of any
         adjustment in the Exercise Price, the number of Warrant Shares issuable
         hereunder upon exercise shall be inversely proportionately increased or
         decreased as the case may be, such that aggregate purchase price for
         Warrant Shares upon full exercise of this Warrant shall remain the
         same.

13.      Voluntary Adjustment by the Company. The Company may at its option, at
         any time during the term of this Warrant, reduce but not increase the
         then current Exercise Price to any amount and for any period of time
         deemed appropriate by the Board of Directors of the Company.

14.      Notice of Adjustment; Notice of Events. (i) Whenever the number of
         Warrant Shares or number or kind of securities or other property
         purchasable upon the exercise of this Warrant or the Exercise Price is
         adjusted, the Company shall promptly mail to the Holder of this Warrant
         a notice setting forth the number of Warrant Shares (and other
         securities or property) purchasable upon the exercise of this Warrant
         and the Exercise Price of such Warrant Shares after such adjustment and
         setting forth the computation of such adjustment and a brief statement
         of the facts requiring such adjustment. (ii) If: (A) the Company shall
         declare a dividend (or any other distribution) on its Common Stock; or
         (B) the Company shall declare a special nonrecurring cash dividend on
         or a redemption of its Common Stock; or (C) the Company shall authorize
         the granting to all holders of the Common Stock rights or warrants to
         subscribe for or purchase any shares of capital stock of any class or
         of any rights; or (D) the approval of any stockholders of the Company
         shall be required in connection with any reclassification of the Common
         Stock of the Company, any consolidation or merger to which the Company
         is a party, any sale or transfer of all or substantially all of the
         assets of the Company, or any compulsory share exchange whereby the
         Common Stock is converted into other securities, cash or property; or
         (E) the Company shall authorize the voluntary dissolution, liquidation
         or winding up of the affairs of the Company, then the Company shall
         cause to be mailed to each Warrant holder at their last addresses as
         they shall appear upon the Warrant register of the Company, at least 30
         calendar days prior to the applicable record or effective date
         hereinafter specified, a notice stating (x) the date on which a record
         is to be taken for the purpose of such dividend, distribution,
         redemption, rights or warrants, or if a record is not to be taken, the
         date as of which the holders of Common Stock of record to be entitled
         to such dividend, distributions, redemption, rights or warrants are to
         be determined or (y) the date on which such reclassification,
         consolidation, merger, sale, transfer or share exchange is expected to
         become effective or close, and the date as of which it is expected that
         holders of Common Stock of record shall be entitled to exchange their
         shares of Common Stock for securities, cash or other property
         deliverable upon such


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         reclassification, consolidation, merger, sale, transfer, share
         exchange, dissolution, liquidation or winding up.

15.      Authorized Shares. The Company covenants that it will at all times
         reserve from its authorized and unissued Common Stock a sufficient
         number of shares, which shall not be less than the number required by
         Section 3.10 of the Purchase Agreement, for the issuance of shares of
         Common Stock upon the exercise of any and all purchase rights under
         this Warrant (and/or the conversion of the Debentures). The Company
         further covenants that its issuance of this Warrant shall constitute
         full authority to its officers who are charged with the duty of
         executing stock certificates to execute and issue the necessary
         certificates for the Warrant Shares upon the exercise of the purchase
         rights under this Warrant. The Company will take all such reasonable
         action as may be necessary to assure that such Warrant Shares may be
         issued as provided herein without violation of any applicable law,
         regulation, or rule of any applicable market or exchange.

16.      9.99% Limitation.

(a)      Notwithstanding anything to the contrary contained herein, the number
         of shares of Common Stock that may be acquired by a Holder upon
         exercise pursuant to the terms hereof at any time shall not exceed a
         number that, when added to the total number of shares of Common Stock
         deemed beneficially owned at such time by such Holder (other than by
         virtue of the ownership of securities or rights to acquire securities
         (including the Debentures and Warrants) that have limitations on the
         Holder's right to convert, exercise or purchase similar to the
         limitation set forth herein), together with all shares of Common Stock
         deemed beneficially owned at such time (other than by virtue of the
         ownership of securities or rights to acquire securities that have
         limitations on the right to convert, exercise or purchase similar to
         the limitation set forth herein) by the Holder's "affiliates" (as
         defined in Rule 144 of the Act) ("AGGREGATION PARTIES") that would be
         aggregated for purposes of determining whether a group under Section
         13(d) of the Securities Exchange Act of 1934 as amended, exists, would
         exceed 9.99% of the total issued and outstanding shares of the Common
         Stock (the "RESTRICTED OWNERSHIP PERCENTAGE"). Each Holder shall have
         the right (w) at any time and from time to time to reduce its
         Restricted Ownership Percentage immediately upon notice to the Company
         and (x) (subject to waiver) at any time and from time to time, to
         increase its Restricted Ownership Percentage immediately upon notice to
         the Company in the event of the announcement as pending or planned, of
         a transaction or event referred to in Section 4 or Section 18(g) or (k)
         of the Debentures. For this purpose, any material modification of the
         terms of a Change in Control Transaction (as defined in the Debentures)
         will be deemed to result in a new Change in Control Transaction. The
         Company shall provide all Holders with the later of (i) 20 days' prior
         written notice of any such Change in Control Transaction, to the extent
         the Company has prior knowledge of a Change in Control Transaction; or
         (ii) notice on the day immediately following the Company's learning of
         any such transaction, but only after, in the case of (i) and (ii), such
         Change in Control Transaction has been publicly disclosed.


                                       10
<PAGE>   11
(b)      The Holder covenants at all times on each day (each such day being
         referred to as a "COVENANT DAY") as follows: During the balance of such
         Covenant Day and the succeeding sixty-one (61) days (the balance of
         such Covenant Day and the succeeding 61 days being referred to as the
         "COVENANT PERIOD") such Holder will not acquire shares of Common Stock
         pursuant to any right (including conversion of Debentures and exercise
         of Warrants) existing at the commencement of the Covenant Period to the
         extent the number of shares so acquired by such Holder and its
         Aggregation Parties (ignoring all dispositions) would exceed:

                           (x) the Restricted Ownership Percentage of the total
                  number of shares of Common Stock outstanding at the
                  commencement of the Covenant Period,

                           minus

                           (y) the number of shares of Common Stock actually
                  owned by such Holder and its Aggregation Parties at the
                  commencement of the Covenant Period.

         A new and independent covenant will be deemed to be given by the Holder
         as of each moment of each Covenant Day. No covenant will terminate,
         diminish or modify any other covenant. The Holder agrees to comply with
         each such covenant. This Section 16 controls in the case of any
         conflict with any other provision of the Purchase Agreement or related
         documents.

(c)      The Company's obligation to issue shares of Common Stock which would
         exceed such limits referred to in this Section 16 shall be suspended to
         the extent necessary until such time, if any, as shares of Common Stock
         may be issued in compliance with such restrictions.

17.      Compliance with Securities Laws.

(a)      The Holder hereof acknowledges that the Warrant Shares acquired upon
         the exercise of this Warrant, if not registered (or if no exemption
         from registration exists), will have restrictions upon resale imposed
         by state and federal securities laws. Each certificate representing the
         Warrant Shares issued to the Holder upon exercise (if not registered,
         for resale or otherwise, or if no exemption from registration exists)
         will bear substantially the following legend:

         THE SHARES REPRESENTED BY THIS CERTIFICATE HAVE NOT BEEN REGISTERED
         WITH THE SECURITIES AND EXCHANGE COMMISSION OR THE SECURITIES
         COMMISSION OF ANY STATE IN RELIANCE UPON AN EXEMPTION FROM REGISTRATION
         UNDER THE SECURITIES ACT OF 1933, AS AMENDED (THE "SECURITIES ACT"),
         AND, ACCORDINGLY, MAY NOT BE OFFERED, TRANSFERRED, SOLD OR OTHERWISE
         DISPOSED OF EXCEPT PURSUANT TO AN EFFECTIVE REGISTRATION STATEMENT
         UNDER THE SECURITIES ACT OR PURSUANT TO AN AVAILABLE EXEMPTION FROM, OR
         IN A TRANSACTION NOT SUBJECT TO, THE REGISTRATION REQUIREMENTS


                                       11
<PAGE>   12
         OF THE SECURITIES ACT AND IN ACCORDANCE WITH APPLICABLE STATE
         SECURITIES LAWS.

(b)      Without limiting the Investor's right to transfer, assign or otherwise
         convey the Warrant or Warrant Shares in compliance with all applicable
         securities laws, the Investor of this Warrant, by acceptance hereof,
         acknowledges that this Warrant and the Warrant Shares to be issued upon
         exercise hereof are being acquired solely for the Investor's own
         account and not as a nominee for any other party, and that the Investor
         will not offer, sell or otherwise dispose of this Warrant or any
         Warrant Shares to be issued upon exercise hereof except under
         circumstances that will not result in a violation of applicable federal
         and state securities laws.

18.      Miscellaneous.

(a)      ISSUE DATE; CHOICE OF LAW; VENUE; JURISDICTION. THE PROVISIONS OF THIS
         WARRANT SHALL BE CONSTRUED AND SHALL BE GIVEN EFFECT IN ALL RESPECTS AS
         IF IT HAD BEEN ISSUED AND DELIVERED BY THE COMPANY ON THE DATE HEREOF.
         THIS WARRANT SHALL BE BINDING UPON ANY SUCCESSORS OR ASSIGNS OF THE
         COMPANY. THIS WARRANT WILL BE CONSTRUED AND ENFORCED IN ACCORDANCE WITH
         AND GOVERNED BY THE LAWS OF THE STATE OF NEW YORK, EXCEPT FOR MATTERS
         ARISING UNDER THE ACT, WITHOUT REFERENCE TO PRINCIPLES OF CONFLICTS OF
         LAW. EACH OF THE PARTIES CONSENTS TO THE EXCLUSIVE JURISDICTION OF THE
         FEDERAL AND STATE COURTS SITTING IN THE COUNTY OF NEW YORK IN THE STATE
         OF NEW YORK IN CONNECTION WITH ANY DISPUTE ARISING UNDER THIS WARRANT
         AND HEREBY WAIVES, TO THE MAXIMUM EXTENT PERMITTED BY LAW, ANY
         OBJECTION, INCLUDING ANY OBJECTION BASED ON FORUM NON CONVENIENS OR
         VENUE, TO THE BRINGING OF ANY SUCH PROCEEDING IN SUCH JURISDICTION.
         EACH PARTY HEREBY AGREES THAT IF THE OTHER PARTY TO THIS WARRANT
         OBTAINS A JUDGMENT AGAINST IT IN SUCH A PROCEEDING, THE PARTY WHICH
         OBTAINED SUCH JUDGMENT MAY ENFORCE SAME BY SUMMARY JUDGMENT IN THE
         COURTS OF ANY COUNTRY HAVING JURISDICTION OVER THE PARTY AGAINST WHOM
         SUCH JUDGMENT WAS OBTAINED, AND EACH PARTY HEREBY WAIVES ANY DEFENSES
         AVAILABLE TO IT UNDER LOCAL LAW AND AGREES TO THE ENFORCEMENT OF SUCH A
         JUDGMENT. EACH PARTY TO THIS WARRANT IRREVOCABLY CONSENTS TO THE
         SERVICE OF PROCESS IN ANY SUCH PROCEEDING BY THE MAILING OF COPIES
         THEREOF BY REGISTERED OR CERTIFIED MAIL, POSTAGE PREPAID, TO SUCH PARTY
         AT ITS ADDRESS IN ACCORDANCE WITH SECTION 18(C). NOTHING HEREIN SHALL
         AFFECT THE RIGHT OF ANY PARTY TO SERVE PROCESS IN ANY OTHER MANNER
         PERMITTED BY LAW.

(b)      Modification and Waiver. This Warrant and any provisions hereof may be
         changed, waived, discharged or terminated only by an instrument in
         writing signed by the party


                                       12
<PAGE>   13
         against which enforcement of the same is sought. Any amendment effected
         in accordance with this paragraph shall be binding upon the Investor,
         each future Holder of this Warrant and the Company. No waivers of, or
         exceptions to, any term, condition or provision of this Warrant, in any
         one or more instances, shall be deemed to be, or construed as, a
         further or continuing waiver of any such term, condition or provision.

(c)      Notices. Any notice, request or other document required or permitted to
         be given or delivered to the Investor or future Holders hereof or the
         Company shall be personally delivered or shall be sent by certified or
         registered mail, postage prepaid, to the Investor or each such Holder
         or the Company, as the case may be, as provided for in the Purchase
         Agreement or as otherwise instructed in writing by such person. All
         notices under this Warrant shall be deemed to have been given when
         received.

         A party may from time to time change the address to which notices to it
         are to be delivered or mailed hereunder by notice given in accordance
         with the provisions of this Section 18(c).

(d)      Severability. Whenever possible, each provision of this Warrant shall
         be interpreted in such manner as to be effective and valid under
         applicable law, but if any provision of this Warrant is held to be
         invalid, illegal or unenforceable in any respect under any applicable
         law or rule in any jurisdiction, such invalidity, illegality or
         unenforceability shall not affect the validity, legality or
         enforceability of any other provision of this Warrant in such
         jurisdiction or affect the validity, legality or enforceability of any
         provision in any other jurisdiction, but this Warrant shall be
         reformed, construed and enforced in such jurisdiction as if such
         invalid, illegal or unenforceable provision had never been contained
         herein.

(e)      No Impairment. The Company will not, by amendment of its Certificate of
         Incorporation or through any reorganization, transfer of assets,
         consolidation, merger, dissolution, issue or sale of securities or any
         other voluntary action, 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 Holder against impairment. Without
         limiting the generality of the foregoing, the Company (a) will not
         increase the par value of any Warrant Shares above the amount payable
         therefor on such exercise, and (b) will take all such action as may be
         reasonably necessary or appropriate in order that the Company may
         validly and legally issue fully paid and nonassessable Warrant Shares
         on the exercise of this Warrant.


                                       13
<PAGE>   14
         IN WITNESS WHEREOF, the Company has caused this Warrant to be executed
by its officers thereunto duly authorized.


                                        APPLIEDTHEORY CORPORATION

                                        By: /s/ David A Buckel
                                              Name:  David A Buckel
                                              Title: Senior Vice President / CFO


ATTEST:



/s/ Robert F. Mechur, Esq.


                                       14
<PAGE>   15
                               NOTICE OF EXERCISE



To:      APPLIEDTHEORY CORPORATION

(1)      The undersigned hereby elects to exercise its Warrant for and to
         purchase thereunder, ______ shares of Common Stock, [and herewith makes
         payment therefor of $_______] or [and elects to utilize the cashless
         exercise provisions of Section 3(b) of this Warrant, resulting in
         ______ shares of Common Stock issuable hereunder].

(2)      Please issue a certificate or certificates representing said shares of
         Common Stock in the name of the undersigned or in such other name as is
         specified below:

                           -------------------------------
                           (Name)

                           -------------------------------
                           (Address)
                           -------------------------------

(3)      [THE UNDERSIGNED HEREBY ELECTS TO EFFECT DELIVERY OF THE WARRANT
         PURSUANT TO THE BOOK ENTRY PROVISIONS OF SECTION 3(a)(v) THEREOF] or
         [PLEASE ISSUE A NEW WARRANT FOR THE UNEXERCISED PORTION OF THE ATTACHED
         WARRANT IN THE NAME OF THE UNDERSIGNED OR IN SUCH OTHER NAME AS IS
         SPECIFIED BELOW:

                                        -----------------------------------
                                        (NAME)

--------------------                    -----------------------------------
(DATE)                                  (SIGNATURE)

                                        -----------------------------------
                                        (ADDRESS)]



Dated:


------------------------------
Signature
<PAGE>   16
                                 ASSIGNMENT FORM

                    (To assign the foregoing Warrant, execute
                   this form and supply required information.
                 Do not use this form to exercise the Warrant.)



         FOR VALUE RECEIVED, the foregoing Warrant and all rights evidenced
thereby are hereby assigned to

_______________________________________________ whose address is

_______________________________________________________________.

_______________________________________________________________

                                                          Dated: ______________,


                               Holder's Signature: _____________________________

                                  Holder's Address:_____________________________



Signature Guaranteed:  ___________________________________________




NOTE: The signature to this Assignment Form must correspond with the name as it
appears on the face of the Warrant, without alteration or enlargement or any
change whatsoever, and must be guaranteed by a bank or trust company. Officers
of corporations and those acting in an fiduciary or other representative
capacity should file proper evidence of authority to assign the foregoing
Warrant.
<PAGE>   17
                                 INDEX OF TERMS

<TABLE>
<CAPTION>
                                                                AS DEFINED ON PAGE
                                                                ------------------

<S>                                                             <C>
"AGGREGATION PARTIES"...........................................................10

"COMMON STOCK"...................................................................1

"COMPANY"........................................................................1

"CONVERTIBLE SECURITIES".........................................................8

"COVENANT PERIOD"...............................................................11

"DISTRIBUTED PROPERTY"...........................................................6

"DTC"............................................................................3

"DWAC"...........................................................................3

"EXERCISE PRICE".................................................................1

"FAIR MARKET VALUE"..............................................................7

"FAST"...........................................................................3

"HOLDER".........................................................................2

"INVESTOR".......................................................................1

"MARKET VALUE"...................................................................4

"PURCHASE AGREEMENT".............................................................1

"RESTRICTED OWNERSHIP PERCENTAGE"...............................................10

"SECURITIES ACT"................................................................11

"TERMINATION DATE"...............................................................1

"TRADING DAY"....................................................................2

"WARRANT SHARES".................................................................1
</TABLE>