Document:

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                                                                     EXHIBIT 4.2

        THESE SECURITIES HAVE NOT BEEN REGISTERED UNDER THE SECURITIES ACT OF
        1933, AS AMENDED, OR ANY STATE SECURITIES LAWS. THEY HAVE BEEN ACQUIRED
        SOLELY FOR INVESTMENT AND NOT WITH A VIEW TO, OR IN CONNECTION WITH, THE
        SALE OR DISTRIBUTION THEREOF. THEY MAY NOT BE SOLD, OFFERED FOR SALE,
        PLEDGED, HYPOTHECATED OR OTHERWISE DISTRIBUTED IN THE ABSENCE OF AN
        EFFECTIVE REGISTRATION STATEMENT RELATED THERETO OR AN OPINION OF
        COUNSEL, SATISFACTORY IN FORM AND SUBSTANCE TO THE COMPANY THAT SUCH
        REGISTRATION IS NOT REQUIRED UNDER THE SECURITIES ACT OF 1933, AS
        AMENDED AND ANY APPLICABLE STATE SECURITIES LAWS.

NO. 1                               WARRANT                     FEBRUARY 8, 1999

                  TO PURCHASE 60,000 SHARES OF COMMON STOCK OF
                     UNIVERSAL ACCESS, INC. (THE "COMPANY")

        1. Number of Shares; Exercise Price: Term. This certifies that Broadmark
Capital Corporation or its permitted assigns (the "Holder") is entitled, upon
the terms and subject to the conditions hereinafter set forth, at any time after
the date hereof and at or prior to 11:59 p.m. Central Time, on February 8, 2004
(the "Expiration Time"), but not thereafter, to acquire from the Company, in
whole or in part, from time to time, up to 60,000 fully paid and nonassessable
shares (the "Shares") of Common Stock, no par value per share, of the Company
("Common Stock"), at a purchase price of Three Dollars ($3.00) per Share (the
"Exercise Price"). The number of Shares, type of security for which this Warrant
is exercisable and Exercise Price are subject to adjustment as provided herein,
and all references to "Common Stock" and "Exercise Price" herein shall be deemed
to include any such adjustment or series of adjustments.

        2.     Exercise of Warrant.

               (a) In general, the purchase rights represented by this Warrant
are exercisable by the Holder, in whole or in part, at any time, or from time to
time, prior to the Expiration Time, by the surrender of this Warrant and the
Notice of Exercise annexed hereto, all duly completed and executed on behalf of
the Holder, at the office of the Company in Chicago, Illinois, (or such other
office or agency of the Company as it may designate by notice in writing to the
Holder at the address of the Holder appearing on the books of the Company), and
upon payment of the Exercise Price for the Shares thereby purchased (by cash,
certified or cashier's check, or wire transfer payable to the order of the
Company, at the time of exercise in an amount equal to the purchase price of the
Shares thereby purchased, or by net issue exercise pursuant to Section 2(b)
hereof). Thereupon the Holder shall be entitled to receive from the Company a
stock certificate in proper form representing the number of Shares so purchased,
and a new Warrant in substantially identical form and dated as of such exercise
for the purchase of that number of Shares equal to the difference, if any,
between the number of Shares subject hereto and the number of Shares as to which
this Warrant is so exercised.

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               (b) Net Issue Exercise. Notwithstanding any provisions in this
Warrant to the contrary, if the fair market value of one share of Series A
Preferred Stock is greater than the Exercise Price (at the date of calculation
as set forth below), in lieu of exercising this Warrant by paying the Exercise
Price in cash, the Holder may elect to receive shares equal to the value (as
determined below) of this Warrant (or the portion of the Warrant being exercised
and canceled) by surrender of this Warrant at the principal office of the
Company together with the properly endorsed Notice of Exercise and notice of
such election in which event the Company shall issue to the Holder a number of
shares of Series A Preferred Stock computed using the following formula:

                             X=Y(A-B) / A

        Where         X=     the number of shares of Series A Preferred Stock to
                             be issued to the Holder

                      Y=     the number of shares of Series A Preferred Stock
                             purchasable under the Warrant, or if only a portion
                             of the Warrant is being exercised, the portion of
                             the Warrant being exercised and canceled (at the
                             date of such calculation)

                      A=     the Fair Market Value

                      B=     the Exercise Price

For purposes of the above calculation, Fair Market Value shall be determined by
the Company's Board of Directors in good faith; provided, however, that where
there exists a public market for the Company's Common Stock at the time of such
exercise, the Fair Market Value shall be the average of the closing bid and
asked prices of the Common Stock quoted in the Over-The-Counter Market Summary
or the last reported sale price of the Common Stock or the closing price quoted
on the NASDAQ National Market or on any exchange on which the Common Stock is
listed, whichever is applicable, as published in the Central Edition of The Wall
Street Journal for the five (5) trading days prior to the date of determination
of Fair Market Value. Notwithstanding the foregoing, if the Warrant is exercised
in connection with the Company's initial public offering of Common Stock, the
Fair Market Value shall be the per share offering price to the public of the
Company's initial public offering.

        3. Issuance of Shares. Certificates for Shares purchased hereunder shall
be delivered to the Holder within a reasonable time after the date on which this
Warrant shall have been exercised in accordance with the terms hereof. All
Shares that may be issued upon the exercise of this Warrant shall, upon such
exercise, be duly and validly authorized and issued, fully paid and
nonassessable and free from all taxes, liens and charges in respect of the
issuance thereof (other than liens or charges created by or imposed upon the
Holder or taxes in respect of any transfer occurring contemporaneously or
otherwise specified herein). The Company agrees that the Shares so issued

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shall be and shall for all purposes be deemed to have been issued to the 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 or converted in accordance with the
terms hereof.

        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 any fractional Share to which the Holder would otherwise be
entitled, the Holder shall be entitled, at its option, to receive either (i) a
cash payment equal to the excess of fair market value for such fractional Share
above the Exercise Price for such fractional share (as determined in good faith
by the Company) or (ii) a whole Share if the Holder tenders the Exercise Price
for one whole share.

        5. No Rights as Shareholders. This Warrant does not entitle the Holder
to any voting rights or other rights as a shareholder of the Company prior to
the exercise hereof.

        6. Charges, Taxes and Expenses. Certificates for Shares issued upon
exercise of this Warrant shall be issued in the name of the Holder. Issuance of
certificates for Shares upon the exercise of this Warrant shall be made without
charge to the Holder for any issue or transfer tax or other incidental expense
in respect of the issuance of such certificates, all of which taxes and expenses
shall be paid by the Company.

        7. Exchange and Registry of Warrant. This Warrant is exchangeable, upon
the surrender hereof by the registered Holder at the office or agency of the
Company referenced in Section 2 above, for a new Warrant on substantially
identical form and dated as of such exchange. The Company shall maintain at the
office or agency referenced in Section 2 above, a registry showing the name and
address of the registered Holder of this Warrant.

        This Warrant may be surrendered for exchange or exercise, in accordance
with its terms, at the office of the Company, and the Company shall be entitled
to rely in all respects, prior to written notice to the contrary, upon such
registry.

        8. Loss, Theft, Destruction or Mutilation of Warrant. Upon receipt by
the Company of evidence reasonably satisfactory to it of the loss, theft,
destruction or mutilation of this Warrant, and in the case of loss, theft or
destruction, of indemnity or security reasonably satisfactory to it, and upon
reimbursement to the Company of all reasonable expenses incidental thereto, and
upon surrender and cancellation of this Warrant, if mutilated, the Company will
make and deliver a new Warrant of like tenor and dated as of such cancellation
and reissuance, in lieu of this Warrant.

        9. 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 or a Sunday or shall be a legal holiday, then such
action may be taken or such right may be exercised on the next succeeding day
not a Saturday or a Sunday or a legal holiday.

        10. Adjustments of Rights. The purchase price per Share and the number
of Shares purchasable hereunder are subject to adjustment from time to time as
follows:

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               (a) Merger or Consolidation. If at any time there shall be a
merger or a consolidation of the Company with or into another corporation when
the Company is not the surviving corporation, then, as a condition of such
merger or consolidation, lawful provision shall be made so that the Holder shall
thereafter be entitled to receive upon exercise of this Warrant, during the
period specified herein and upon payment of the aggregate Exercise Price then in
effect, the number of shares of stock or other securities or property (including
cash) of the successor corporation resulting from such merger or consolidation,
to which the holder of the stock deliverable upon exercise of this Warrant would
have been entitled in such merger or consolidation if this Warrant had been
exercised immediately before such merger or consolidation. In any such case,
appropriate adjustment shall be made in the application of the provisions of
this Warrant with respect to the rights and interests of the Holder after the
merger or consolidation. This provision shall apply to successive mergers or
consolidations.

               (b) Reclassification, Recapitalization, etc. If the Company at
any time shall, by subdivision, combination or reclassification of securities,
recapitalization, automatic conversion, or other similar event affecting the
number or character of outstanding Shares, or otherwise, change any 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, this Warrant
shall thereafter represent the right to acquire such number and kind of
securities as would have been issuable as the result of such change with respect
to the securities that were subject to the purchase rights under this Warrant
immediately prior to such subdivision, combination, reclassification or other
change.

               (c) Split, Subdivision or Combination of Shares. If the Company
at any time while this Warrant remains outstanding and unexpired shall split,
subdivide or combine the securities as to which purchase rights under this
Warrant exist, the Exercise Price shall be proportionately decreased in the case
of a split or subdivision or proportionately increased in the case of a
combination.

               (d) Dividends. If the Company at any time while this Warrant is
outstanding and unexpired shall pay a dividend with respect to Common Stock
payable solely in shares of the Company's equity securities, then the Exercise
Price shall be adjusted, from and after the date of determination of the
shareholders entitled to receive such dividend, to that price determined by
multiplying the Exercise Price in effect immediately prior to such date of
determination by a fraction (i) the numerator of which shall be the total number
of shares of Common Stock outstanding immediately prior to such dividend, and
(ii) the denominator of which shall be the total number of shares of Common
Stock outstanding immediately after such dividend.

        The number of shares of Common Stock at any time outstanding (A) shall
include, in addition to outstanding shares of Common Stock, the number of shares
of Common Stock into which the Series B Cumulative Convertible Preferred Stock,
the Series A Cumulative Convertible Preferred Stock, the Series C Convertible
Preferred Stock, or any of the Company's other equity securities, indebtedness
or other securities are convertible; (B) shall include the number of shares of
Common Stock into which any of the Company's vested options or warrants
(including warrants exercisable

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for equity securities or indebtedness convertible into Common Stock) are then
convertible; and (C) shall not include treasury shares. This paragraph shall
apply only if and to the extent that, at the time of such event, this Warrant is
then exercisable for Common Stock.

               (e) Adjustment of Number of Shares. Upon each adjustment in the
Exercise Price pursuant to 10(c) or 10(d) hereof, the number of Shares
purchasable hereunder shall be adjusted, to the nearest whole Share, to the
product obtained by multiplying the number of Shares purchasable immediately
prior to such adjustment in the Exercise Price by a fraction (i) the numerator
of which shall be the Exercise Price immediately prior to such adjustment, and
(ii) the denominator of which shall be the Exercise Price immediately after such
adjustment.

               (f) Dividends. If the Company declares or pays a dividend or
other distribution upon the Common Stock other than a stock dividend payable
solely in shares of the Company's equity securities (a "Dividend"), then the
Company shall pay to the Holder at the time of payment thereof the Dividend
which would have been paid to such Holder if such Holder had held the maximum
number of Shares acquirable upon complete exercise of the outstanding portion of
this Warrant immediately prior to the date on which a record is taken for such
Dividend, or, if no record is taken, the date as of which the record holders of
Common Stock entitled to such dividends are to be determined.

        11. Stock to Be Reserved. The Company will at all times reserve and keep
available out of its authorized but unissued Common Stock, solely for the
purpose of issue upon the exercise of it's Warrant as herein provided, such
number of shares of Common Stock as shall be issuable upon Exercise of this
Warrant (the "Reserved Shares"). The Company covenants that, when the Reserved
Shares are issued in accordance with the terms hereof (including payment of the
Exercise Price), the Reserved Shares shall be duly and validly issued and fully
paid and nonassessable and free from all taxes, liens and charges with respect
to the issue thereof. Without limiting the generality of the foregoing, the
Company covenants that it will from time to time take all such action as may be
reasonably necessary to assure that the par value per share of the Common Stock
at all times shall be equal to or less than the effective Exercise Price.

        The Company will take all such action as may be reasonably necessary to
assure that all Reserved Shares may be issued without violation of any
applicable law or regulation, or of any requirements of any national securities
exchange upon which the Common Shares of the Company may be listed. The Company
will not take any action which results in any adjustment of the Exercise Price
if the total number of shares of Common Stock issued and issuable after such
action upon exercise of this Warrant would exceed the total number of shares of
Common Stock then authorized by the Company's Articles of Incorporation. The
Company has not granted and will not grant any right of first refusal or similar
right with respect to Shares issuable upon exercise of this Warrant, and there
are no preemptive rights associated with such Shares.

        12. Issue Tax. The issuance of certificates for shares of Common Stock
upon exercise of this Warrant shall be made without charge to the Holder for any
issuance or transfer tax in respect

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thereof; provided that the Company shall not be required to pay any tax which
may be payable in respect of any transfer involved in the issuance and delivery
of any certificate in a name other than that of the Holder.

        13. Closing of Books. The Company will at no time close its transfer
books against the transfer of the shares of Common Stock issued or issuable upon
the exercise of this Warrant in any manner which interferes with the timely
exercise of this Warrant.

        14. Notices of Record Dates. If the Company sets a record date for
holders of any class of securities for the purpose of determining the holders
entitled to receive any dividend or other distribution (other than cash
dividends out of earned surplus), or any right to subscribe for, purchase or
otherwise acquire any shares of stock of any class or any other securities or
property, or to receive any other right, then at least twenty (20) days, or such
shorter period as is practicable but in no event less than ten (10) days, before
any such record date, the Company shall give notice (the "Notice") to the Holder
specifying the date on which such record is to be taken. The Notice shall state
whether the record date is subject to the effectiveness of a registration
statement under the Securities Act of 1933, as amended (the "Securities Act") or
to a favorable vote of stockholders, if either is required.

        15. Warrant Transferrable. Subject to compliance with applicable state
and federal securities laws, this Warrant and the Holder's rights hereunder are
transferrable, in whole or in part, without charge to the Holder upon surrender
of this Warrant with a properly executed assignment, in form and substance
reasonably acceptable to the Company.

        16. Notice of Adjustments; Notices. Whenever the Exercise Price or
number or type of securities issuable hereunder shall be adjusted pursuant to
Section 10 hereof, the Company shall issue and provide to the Holder a
certificate signed by an officer of the Company setting forth, in reasonable
detail, the event requiring the adjustment, the amount of the adjustment, the
method by which such adjustment was calculated and the Exercise Price and number
of Shares purchasable hereunder after giving effect to such adjustment.

        17. Governing Law. This Warrant shall be binding upon any successors or
assigns of the Company. This Warrant shall constitute a contract under the laws
of Illinois and for all purposes shall be construed in accordance with and
governed by the laws of said state, without giving effect to the conflict of
laws principles.

        18. Attorneys' Fees. In any litigation, arbitration or court proceeding
between the Company and the Holder relating hereto, the prevailing party shall
be entitled to reasonable attorneys' fees and expenses incurred in enforcing
this Warrant.

        19. Amendments. This Warrant may be amended and the observance of any
term of this Warrant may be waived only with the written consent of the Company
and the Holder.

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        20. Notice. Except as otherwise specified herein, any notice, demand or
request required or permitted to be given pursuant to the terms of this
Agreement shall be in writing and shall be deemed given (i) when delivered
personally or by verifiable facsimile transmission (with a hard copy to follow)
on or before 5:00 p.m., central time, on a business day or, if the day is not a
business day, on the next succeeding business day, (ii) on the next business day
after timely delivery to an overnight courier and (iii) on the third business
day after deposit in the U.S. mail (certified or registered mail, return receipt
requested, postage prepaid), addressed as follows:

               If to the Company:

               Universal Access, Inc.
               100 North Riverside Plaza
               Suite 2200 Chicago, IL 60606
               Phone: 312-660-5000
               Fax: 312-660-5050

               With a copy to:

               Shefsky & Froelich Ltd.
               444 North Michigan Avenue
               Suite 2500
               Chicago, Illinois 60611
               Attn: Mitchell D. Goldsmith, Esq.
               Phone: 312-836-4006
               Fax: 312-527-3194

and if to any Holder, at its address as shown on the stock records of the
Company or such other address as any such party shall deliver to the Company.

        21. Entire Agreement. This Warrant and the Notice of Exercise attached
hereto contain the entire agreement between the parties with respect to the
subject matter hereof and supersede all prior and contemporaneous arrangements
or undertakings with respect thereto.

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        IN WITNESS WHEREOF, Universal Access, Inc. has caused this Warrant to be
executed by its duly authorized officer.

Dated: February 8, 1999

                                        UNIVERSAL ACCESS, INC.

                                        By: /s/ PATRICK SHUTT
                                           ---------------------
                                        Name: Patrick Shutt
                                        Title: President

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                               NOTICE OF EXERCISE

To:     Universal Access, Inc.

        1. The undersigned hereby elects to purchase ____________________ shares
(the "Shares") of Common Stock, no par value per share, of Universal Access,
Inc. (the "Company") pursuant to the terms of the attached Warrant, and tenders
herewith payment of the purchase price.

        2. The Shares to be received by the undersigned upon exercise of the
Warrant are being acquired for its own account, not as a nominee or agent, and
not with a view to resale or distribution of any part thereof, and the
undersigned has no present intention of selling, granting any participation in,
or otherwise distributing the same, except in compliance with applicable federal
and state securities laws. The undersigned further represents that it does not
have any contract, undertaking, agreement or arrangement with any person to
sell, transfer or grant participation to such person or to any third person,
with respect to the Shares. The undersigned has received all the information it
considers necessary or appropriate for deciding whether to purchase the Shares.

        3. The undersigned understands that the Shares are characterized as
"restricted securities" under the federal securities laws inasmuch as they are
being acquired from the Company in transactions not involving a public offering
and that under such laws and applicable regulations such securities may be
resold without registration under the Securities Act of 1933, as amended (the
"Act"), only in certain limited circumstances. In this connection, the
undersigned represents that it is familiar with Rule 144 of the Act, as
presently in effect, and understands the resale limitations imposed thereby and
by the Act.

        4. The undersigned understands the certificates evidencing the Shares
may bear one or all of the following legends:

               (a) "THESE SECURITIES HAVE NOT BEEN REGISTERED UNDER THE
SECURITIES ACT OF 1933. THEY MAY NOT BE SOLD, OFFERED FOR SALE, PLEDGED OR
HYPOTHECATED IN THE ABSENCE OF A REGISTRATION STATEMENT IN EFFECT WITH RESPECT
TO THE SECURITIES UNDER SUCH ACT OR AN OPINION OF COUNSEL SATISFACTORY TO THE
COMPANY THAT SUCH REGISTRATION IS NOT REQUIRED OR UNLESS SOLD PURSUANT TO RULE
144 OF SUCH ACT."

               (b) Any legend required by applicable state law.

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        5. Please issue a certificate or certificates representing said Shares
in the name of the undersigned.

                                             [NAME OF HOLDER]

                                             By:
                                                --------------------------------
Date:                                        Title:
      -------------------                          -----------------------------

        6. Please issue a new Warrant for the unexercised portion of the
attached Warrant in the name of the undersigned.

                                                   [NAME OF HOLDER]

                                             By:
                                                --------------------------------
Date:                                        Title:
      -------------------                          -----------------------------

                                       2<PAGE>   1
                                                                     EXHIBIT 4.3

     THESE SECURITIES HAVE NOT BEEN REGISTERED UNDER THE SECURITIES ACT OF
1933, AS AMENDED, OR ANY STATE SECURITIES LAWS. THEY HAVE BEEN ACQUIRED SOLELY
FOR INVESTMENT AND NOT WITH A VIEW TO, OR IN CONNECTION WITH, THE SALE OR
DISTRIBUTION THEREOF. THEY MAY NOT BE SOLD, OFFERED FOR SALE, PLEDGED,
HYPOTHECATED OR OTHERWISE DISTRIBUTED IN THE ABSENCE OF AN EFFECTIVE
REGISTRATION STATEMENT RELATED THERETO OR AN OPINION OF COUNSEL, SATISFACTORY
IN FORM AND SUBSTANCE TO THE COMPANY THAT SUCH REGISTRATION IS NOT REQUIRED
UNDER THE SECURITIES ACT OF 1933, AS AMENDED AND ANY APPLICABLE STATE
SECURITIES LAWS.

No. A-2                         WARRANT                            March 8, 1999

To Purchase 43,900 Shares of Series A Cumulative Convertible Preferred Stock of
                     Universal Access, Inc. (the "Company")

     1.  Number of Shares; Exercise Price; Term. This certifies that Broadmark
Capital Corporation or its permitted assigns (the "Holder") is entitled, upon
the terms and subject to the conditions hereinafter set forth, at any time
after the date hereof and at or prior to 11:59 p.m. Central Time, on March 8,
2004 (the "Expiration Time"), but not thereafter, to acquire from the Company,
in whole or in part, from time to time, up to 43,900 fully paid and
nonassessable shares (the "Shares") of Series A Cumulative Convertible
Preferred Stock, stated value $3.00 per share, of the Company ("Series A
Preferred Stock"), at a purchase price of $3.00 per Share (the "Exercise
Price"). The number of Shares, type of security for which this Warrant is
exercisable and Exercise Price are subject to adjustment as provided herein,
and all references to "Series A Preferred Stock" and "Exercise Price" herein
shall be deemed to include any such adjustment or series of adjustments.

     2.  Exercise of Warrant.

     (a)  In general, the purchase rights represented by this Warrant are
exercisable by the Holder, in whole or in part, at any time, or from time to
time, prior to the Expiration Time, by the surrender of this Warrant and the
Notice of Exercise annexed hereto, all duly completed and executed on behalf of
the Holder, at the office of the Company in Chicago, Illinois, (or such other
office or agency of the Company as it may designate by notice in writing to the
Holder at the address of the Holder appearing on the books of the Company), and
upon payment of the Exercise Price for the Shares thereby purchased (by cash,
certified or cashier's check, or wire transfer payable to the order of the
Company, at the time of exercise in an amount equal to the purchase price of
the Shares thereby purchased, or by net issue exercise pursuant to Section 2(b)
hereof). Thereupon the Holder shall be entitled to receive from the Company a
stock certificate in proper form representing the number of Shares so
purchased, and a new Warrant in substantially identical form and dated as of
such exercise for the purchase of that number of Shares equal to the
difference, if any, between the number of Shares subject hereto and the number
of Shares as to which this Warrant is so exercised.
<PAGE>   2
      (b)   Net Issue Exercise.  Notwithstanding any provisions in this Warrant
to the contrary, if the fair market value of one share of Series A Preferred
Stock is greater than the Exercise Price (at the date of calculation as set
forth below), in lieu of exercising this Warrant by paying the Exercise Price
in cash, the Holder may elect to receive shares equal to the value (as
determined below) of this Warrant (or the portion of the Warrant being
exercised and canceled) by surrender of this Warrant at the principal office of
the Company together with the properly endorsed Notice of Exercise and notice
of such election in which event the Company shall issue to the Holder a number
of shares of Series A Preferred Stock computed using the following formula:

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

            Where       X =  the number of shares of Series A Preferred Stock
                             to be issued to the Holder

                        Y =  the number of shares of Series A Preferred Stock
                             purchasable under the Warrant, or if only a
                             portion of the Warrant is being exercised, the
                             portion of the Warrant being exercised and
                             canceled (at the date of such calculation)

                        A =  the Fair Market Value

                        B =  the Exercise Price

For purposes of the above calculation, the Fair Market Value shall be determined
by the Company's Board of Directors in good faith; provided, however, that where
there exists a public market for the Company's common shares, no par value per
share (the "Common Shares") at the time of such exercise, the Fair Market Value
shall be the product of (i) the average of the closing bid and asked prices of
the Common shares quoted in the Over-The-Counter Market Summary or the last
reported sale price of the Common Shares or the closing price quoted on the
NASDAQ National Market or on any exchange on which the Common Shares are listed,
whichever is applicable, as published in the Central Edition of The Wall Street
Journal for the five (5) trading days prior to the date of determination of Fair
Market Value and (ii) the number of Common Shares into which each share of
Series A Preferred Stock is convertible at the time of such exercise.
Notwithstanding the foregoing, if the Warrant is exercised in connection with
the Company's initial public offering of Common Shares, the Fair Market Value
shall be the product of (i) the per share offering price to the public of the
Company's initial public offering, and (ii) the number of Common Shares into
which each shares of Series A Preferred Stock is convertible at the time of such
exercise.

      3.    Issuance of Shares.  Certificates for Shares purchased hereunder
shall be delivered to the Holder within a reasonable time after the date on
which this Warrant shall have been exercised in accordance with the terms
hereof. All Shares that may be issue upon the exercise of this Warrant shall,
upon such exercise, be duly and validly authorized and issued, fully paid and
nonassessable

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and free from all taxes, liens and charges in respect of the issuance thereof
(other than liens or charges created by or imposed upon the Holder or taxes in
respect of any transfer occurring contemporaneously or otherwise specified
herein). The Company agrees that the Shares so issued shall be and shall for
all purposes be deemed to have been issued to the 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 or converted in accordance with the terms hereof.

     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 any fractional Share to which the Holder would otherwise
be entitled, the Holder shall be entitled, at its option, to receive either (i)
a cash payment equal to the excess of fair market value for such fractional
Share above the Exercise Price for such fractional share (as determined in good
faith by the Company) or (ii) a whole Share if the Holder tenders the Exercise
Price for one whole share.

     5.   No Rights as Shareholders. This Warrant does not entitle the Holder
to any voting rights or other rights as a shareholder of the Company prior to
the exercise hereof.

     6.   Charges, Taxes and Expenses. Certificates for Shares issued upon
exercise of this Warrant shall be issued in the name of the Holder. Issuance of
certificates for Shares upon the exercise of this Warrant shall be made without
charge to the Holder for any issue or transfer tax or other incidental expense
in respect of the issuance of such certificates, all of which taxes and
expenses shall be paid by the Company.

     7.   Exchange and Registry of Warrant. This Warrant is exchangeable, upon
the surrender hereof by the registered Holder at the office or agency of the
Company referenced in Section 2 above, for a new Warrant on substantially
identical form and dated as of such exchange. The Company shall maintain at the
office or agency referenced in Section 2 above, a registry showing the name and
address of the registered Holder of this Warrant. This Warrant may be
surrendered for exchange or exercise, in accordance with its terms, at the
office of the Company, and the Company shall be entitled to rely in all
respects prior to written notice to the contrary, upon such registry.

     8.   Loss, Theft, Destruction or Mutilation of Warrant. Upon receipt by
the Company of evidence reasonably satisfactory to it of the loss, theft,
destruction or mutilation of this Warrant, and in the case of loss, theft or
destruction, of indemnity or security reasonably satisfactory to it, and upon
reimbursement to the Company of all reasonable expenses incidental thereto, and
upon surrender and cancellation of this Warrant, if mutilated, the Company will
make and deliver a new Warrant of like tenor and dated as of such cancellation
and reissuance, in lieu of this Warrant.

     9.   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 or a Sunday or shall be a legal holiday, then such
action may be taken or such right may be exercised on the next succeeding day
not a Saturday or a Sunday or a legal holiday.

                                       3
<PAGE>   4

     10.  Adjustments of Rights. The purchase price per share and the number of
shares purchasable hereunder are subject to adjustment from time to time as
follows:

          (a)  Merger or Consolidation. If at any time there shall be a merger
or a consolidation of the Company with or into another corporation when the
Company is not the surviving corporation, then, as a condition of such merger
or consolidation, lawful provision shall be made so that the Holder shall
thereafter be entitled to receive upon exercise of this Warrant, during the
period specified herein and upon payment of the aggregate Exercise Price then in
effect, the number of shares of stock or other securities or property
(including cash) of the successor corporation resulting from such merger or
consolidation, to which the holder of the stock deliverable upon exercise of
this Warrant would have been entitled in such merger or consolidation if this
Warrant had been exercised immediately before such merger or consolidation. In
any such case, appropriate adjustment shall be made in the application of the
provisions of this Warrant with respect to the rights and interests of the
Holder after the merger or consolidations.

          (b)  Reclassification, Recapitalization, etc. If the Company at any
time shall, by subdivision, combination or reclassification of securities,
recapitalization, automatic conversion, or other similar event affecting the
number or character of outstanding Shares, or otherwise, change any 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, this Warrant
shall thereafter represent the right to acquire such number and kind of
securities as would have been issuable as the result of such change with
respect to the securities that were subject to the purchase rights under this
Warrant immediately prior to such subdivision, combination, reclassification or
other change.

          (c)  Split Subdivision or Combination of Shares. If the Company at
any time while this Warrant remains outstanding and unexpired shall split,
subdivide or combine the securities as to which purchase rights under this
Warrant exist, the Exercise Price shall be proportionately decreased in the
case of a split or subdivision or proportionately increased in the case of a
combination.

          (d)  Stock Dividends. If the Company at any time while this Warrant
is outstanding and unexpired shall pay a dividend with respect to Series A
Preferred Stock payable solely in shares of the Company's equity securities,
then the Exercise Price shall be adjusted, from and after the date of
determination of the shareholders entitled to receive such dividend, to that
price determined by multiplying the Exercise Price in effect immediately prior
to such date of determination by a fraction (i) the numerator of which shall be
the total number of Common Shares outstanding immediately prior to such
dividend, and (ii) the denominator of which shall be the total number of Common
Shares outstanding immediately after such dividend.

     The number of Common Shares at any time outstanding (A) shall include, in
addition to outstanding Common Shares, the number of Common Shares into which
the Series A Preferred

                                       4
<PAGE>   5
Stock, the Series B Cumulative Convertible Preferred Stock, the Series C
Convertible Preferred Stock, or any of the Company's other equity securities,
indebtedness or other securities are convertible; (B) shall include the number
of Common Shares into which any of the Company's vested options or warrants
(including warrants exercisable for equity securities or indebtedness
convertible into Common Shares) are then convertible; and (B) shall not include
treasury shares. This paragraph shall apply only if and to the extent that, at
the time of such event, this Warrant is then exercisable for Preferred Stock.

          (e)  Adjustment of Number of Shares.  Upon each adjustment in the
Exercise Price pursuant to 10(c) or 10(d) hereof, the number of Shares
purchasable hereunder shall be adjusted, to the nearest whole Share, to the
product obtained by multiplying the number of Shares purchasable immediately
prior to such adjustment in the Exercise Price by a fraction (i) the numerator
of which shall be the Exercise Price immediately prior to such adjustment, and
(ii) the denominator of which shall be the Exercise Price immediately after
such adjustment.

          (f)  Dividends.  If the Company declares or pays a dividend or other
distribution upon the Series A Preferred Stock other than a stock dividend
payable solely in shares of the Company's equity securities (a "Dividend"),
then the Company shall pay to the Holder at the time of payment thereof the
Dividend which would have been paid to such Holder if such Holder had held the
maximum number of Shares acquirable upon complete exercise of the outstanding
portion of this Warrant immediately prior to the date on which a record is
taken for such Dividend, or, if no record is taken, the date as of which the
record holders of Series A Preferred Stock entitled to such dividends are to be
determined.

     11.  Stock to Be Reserved.  The Company will at all times reserve and keep
available out of its authorized but unissued Series A Preferred Stock and
Common Shares, solely for the purpose of issue upon the exercise of this
Warrant as herein provided, such number of shares of (a) Series A Preferred
Stock as shall be issuable upon exercise of this Warrant, and (b) Common Shares
as shall be issuable upon conversion of the Series A Preferred Stock issuable
upon exercise of this Warrant (collectively, the "Reserved Shares"). The
Company covenants that, when the Reserved Shares are issued in accordance with
the terms hereof (including payment of the Exercise Price) or the terms of the
Certificate of Designations, Preferences and Rights of the Series A Preferred
Stock (the "Series A Certificate"), as applicable, the Reserved Shares shall be
duly and validly issued and fully paid and nonassessable and free from all
taxes, liens and charges with respect to the issue thereof. Without limiting
the generality of the foregoing, the Company covenants that it will from time
to time take all such action as may be reasonably necessary to assure that the
par value per share (as opposed to the Stated Value) of the Shares at all times
equal to or less than the effective Exercise Price.

     The Company will take all such action as may be reasonably necessary to
assure that all Reserved Shares may be issued without violation of any
applicable law or regulation, or of any requirements of any national securities
exchange upon which the Common Shares of the Company may be listed. The Company
will not take any action which results in any adjustment of the

                                       5
<PAGE>   6

Exercise Price if the total number of shares of Series A Preferred Stock issued
and issuable after such action upon exercise of this Warrant would exceed the
total number of shares of Series A Preferred Stock then authorized by the
Company's Articles of Incorporation. The Company has not granted and will not
grant any right of first refusal or similar right with respect to Shares
issuable upon exercise of this Warrant, and there are no preemptive rights
associated with such Shares.

     12.  Issue Tax. The issuance of certificates for shares of Series A
Preferred Stock upon exercise of this Warrant shall be made without charge to
the Holder for any issuance or transfer tax in respect thereof; provided that
the Company shall not be required to pay any tax which may be payable in
respect of any transfer involved in the issuance and delivery of any
certificate in a name other than that of the Holder.

     13.  Closing of Books. The Company will at no time close its transfer
books against the transfer of the shares of Series A Preferred Stock issued or
issuable upon the exercise of this Warrant in any manner which interferes with
the timely exercise of this Warrant.

     14.  Notices of Record Dates. If:

          (a)  the Company sets a record date for holders of any class of
securities for the purpose of determining the holders entitled to receive any
dividend or other distribution (other than cash dividends out of earned
surplus), or any right to subscribe for, purchase or otherwise acquire any
shares of stock of any class or any other securities or property, or to receive
any other right, or

          (b)  any Liquidation Event (as defined in the Series A Certificate)
occurs,

then at least twenty (20) days, or such shorter period as is practicable but in
no event less than ten (10) days, before any such record date or Liquidation
Event, the Company shall give notice (the "Notice") to the Holder specifying
(i) the date on which such record is to be taken, and (ii) the date on which
such Liquidation Event is to take place, and the time, if any is to be fixed,
as of which the holders of record of Series A Preferred Stock or Common Shares
will be entitled to exchange their shares of Series A Preferred Stock or Common
Shares, as the case may be, for securities or other property deliverable upon
such Liquidation Event. The Notice shall state whether the Liquidation Event or
the record date is subject to the effectiveness of a registration statement
under the Securities Act of 1933, as amended (the "Securities Act") or to a
favorable vote of stockholders, if either is required.

     15.  Warrant Transferable; Registration Rights. Subject to compliance with
(i) applicable state and federal securities laws, and (ii) any transfer
restrictions set forth in the Series A Certificate, the Warrant Share
Registration and Informational Rights Agreement dated as of September 21, 1998
between the Holder and the Company (the "Registration Rights Agreement"), and
the Shareholders Agreement dated as of February 8, 1999 among the Company and
the parties thereto (collectively, the "Transaction Documents"), this Warrant
and the Holder's rights hereunder

                                       6
<PAGE>   7
are transferrable, in whole or in part, without charge to the Holder upon
surrender of this Warrant with a properly executed assignment, in form and
substance reasonably acceptable to the Company.

     For purposes of the Registration Rights Agreement, the shares of Common
Stock (i) acquired or to be acquired by the Holder pursuant to the conversion
of Shares and (ii) which have not been previously sold pursuant to a
registration statement or Rule 144 promulgated under the Securities Act of
1933, as amended, shall be deemed to constitute "Registrable Securities" as
defined in the Registration Rights Agreement and shall be subject to the
registration rights granted to the Holder under the Registration Rights
Agreement.

     16.  Notice of Adjustments. Whenever the Exercise Price or number or type
of securities issuable hereunder shall be adjusted pursuant to Section 10
hereof, the Company shall issue and provide to the Holder a certificate signed
by an officer of the Company setting forth, in reasonable detail, the event
requiring the adjustment, the amount of the adjustment, the method by which
such adjustment was calculated and the Exercise Price and number of Shares
purchasable hereunder after giving effect to such adjustment.

     17.  Governing Law. This Warrant shall be binding upon any successors or
assigns of the Company. This Warrant shall constitute a contract under the laws
of Illinois and for all purposes shall be construed in accordance with and
governed by the laws of said state, without giving effect to the conflict of
laws principles.

     18.  Attorneys' Fees. In any litigation, arbitration or court proceeding
between the Company and the Holder relating hereto, the prevailing party shall
be entitled to reasonable attorneys' fees and expenses incurred in enforcing
this Warrant.

     19.  Amendments. This Warrant may be amended and the observance of any
term of this Warrant may be waived only with the written consent of the Company
and the Holder.

     20.  Notice. Except as otherwise specified herein, any notice, demand or
request required or permitted to be given pursuant to the terms of this
Agreement shall be in writing and shall be deemed given (i) when delivered
personally or by verifiable facsimile transmission (with a hard copy to follow)
on or before 5:00 p.m., central time, on a business day or, if the day is not a
business day, on the next succeeding business day, (ii) on the next business
day after timely delivery to an overnight courier and (iii) on the third
business day after deposit in the U.S. mail (certified or registered mail,
return receipt requested, postage prepaid), addressed as follows:

          If to the Company:

          Universal Access, Inc.
          100 North Riverside Plaza
          Suite 2200

                                       7

<PAGE>   8

            Chicago, IL 60606
            Phone: 312-660-5000
            Fax:   312-660-5050
            With a copy to:

            Shefsky & Froelich Ltd.
            444 North Michigan Avenue
            Suite 2500
            Chicago, Illinois 60611
            Attn: Mitchell D. Goldsmith, Esq.
            Phone: 312-836-4006
            Fax:   312-527-3194

and if to any Holder, at its address as shown on the stock records of the
Company or such other address as any such party shall deliver to the Company.

      21.   Entire Agreement. This Warrant, the Notice of Exercise attached
hereto and the Transaction Documents contain the entire agreement between the
parties with respect to the subject matter hereof and supersede all prior and
contemporaneous arrangements or undertakings with respect thereto.

      IN WITNESS WHEREOF, Universal Access, Inc. has caused this Warrant to be
executed by its duly authorized officer.

Dated: March 8, 1999

                                                UNIVERSAL ACCESS, INC.

                                                By:
                                                   ---------------------------
                                                Name: Patrick Shutt
                                                Title: President

                                       8
<PAGE>   9
                               NOTICE OF EXERCISE

To:   Universal Access, Inc.

      1.    The undersigned hereby elects to purchase _____________ shares (the
"Shares") of Series A Cumulative Convertible Preferred Stock, stated value
$3.00 per share, of Universal Access, Inc. (the "Company") pursuant to the
terms of the attached Warrant, and tenders herewith payment of the purchase
price.

      2.    The Shares to be received by the undersigned upon exercise of the
Warrant are being acquired for its own account, not as a nominee or agent, and
not with a view to resale or distribution of any part thereof, and the
undersigned has no present intention of selling, granting any participation in,
or otherwise distributing the same, except in compliance with applicable
federal and state securities laws. The undersigned further represents that it
does not have any contract, undertaking, agreement or arrangement with any
person to sell, transfer or grant participation to such person or to any third
person, with respect to the Shares. The undersigned has received all the
information it considers necessary or appropriate for deciding whether to
purchase the Shares.

      3.    The undersigned understands that the Shares are characterized as
"restricted securities" under the federal securities laws inasmuch as they are
being acquired from the Company in transactions not involving a public offering
and that under such laws and applicable regulations such securities may be
resold without registration under the Securities Act of 1933, as amended (the
"Act"), only in certain limited circumstances. In this connection, the
undersigned represents that it is familiar with Rule 144 of the Act, as
presently in effect, and understands the resale limitations imposed thereby and
by the Act.

      4.    The undersigned understands the certificates evidencing the Shares
may bear one or all of the following legends:

            (a)   "THESE SECURITIES HAVE NOT BEEN REGISTERED UNDER THE
            SECURITIES ACT OF 1933. THEY MAY NOT BE SOLD, OFFERED FOR SALE,
            PLEDGED OR HYPOTHECATED IN THE ABSENCE OF A REGISTRATION STATEMENT
            IN EFFECT WITH RESPECT TO THE SECURITIES UNDER SUCH ACT OR AN
            OPINION OF COUNSEL SATISFACTORY TO THE COMPANY THAT SUCH
            REGISTRATION IS NOT REQUIRED OR UNLESS SOLD PURSUANT TO RULE 144 OF
            SUCH ACT."

            (b)   Any legend required by applicable state law.
<PAGE>   10

        5.      Please issue a certificate or certificates representing said
Shares in the name of the undersigned

                                        BROADMARK CAPITAL CORPORATION

                                        By:
                                           ------------------------------------
                                        Title:
                                              ---------------------------------
---------------
     Date

        6.      Please issue a new Warrant for the unexercised portion of the
attached Warrant in the name of the undersigned.

                                        BROADMARK CAPITAL CORPORATION

                                        By:
                                           ------------------------------------
                                        Title:
                                              ---------------------------------
---------------
     Date

                                       2

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