<PAGE>

                                                                     Exhibit 4.2


THIS WARRANT HAS NOT BEEN REGISTERED UNDER THE SECURITIES ACT OF 1933, AS
AMENDED, OR ANY STATE SECURITIES LAW, AND MAY NOT BE OFFERED FOR SALE, SOLD OR
TRANSFERRED UNLESS A REGISTRATION STATEMENT UNDER SUCH ACT AND APPLICABLE STATE
SECURITIES LAWS SHALL BE EFFECTIVE WITH RESPECT THERETO, OR AN EXEMPTION FROM
REGISTRATION UNDER THE SECURITIES ACT AND APPLICABLE STATE SECURITIES LAWS IS
AVAILABLE IN CONNECTION WITH SUCH OFFER, SALE OR TRANSFER.



Warrant to Purchase                            Issue Date:  June 28, 2000
256,621 Shares



                       WARRANT TO PURCHASE COMMON STOCK

                                      of

                         PILOT NETWORK SERVICES, INC.


  THIS CERTIFIES that Marshall Capital Management, Inc. or any subsequent holder
hereof (the "Holder"), has the right to purchase from PILOT NETWORK SERVICES,
             ------
INC., a Delaware corporation (the "Company"), up to 256,621 fully paid and
                                   -------
nonassessable shares of the Company's Common Stock, par value $.001 per share
(the "Common Stock"), subject to adjustment as provided herein, at a price equal
      ------------
to the Exercise Price (as defined below), at any time beginning on the date on
which this Warrant is issued (the "Issue Date") and ending at 5:00 p.m., eastern
                                   ----------
time, on the date that is the fifth (5th) anniversary of the Issue Date (the

"Expiration Date").  This Warrant is issued, and all rights hereunder shall be,
----------------
subject to all of the conditions, limitations and provisions set forth herein
and in the Securities Purchase Agreement of even date herewith by and among the
Company and the Purchasers named therein (the "Securities Purchase Agreement").
                                               -----------------------------
Capitalized terms used herein and not otherwise defined shall have the
respective meanings set forth in the Securities Purchase Agreement or the
Certificate of Designation relating to the Series A Convertible Preferred Stock
issued by the Company pursuant to the Securities Purchase Agreement

("Certificate of Designation").
----------------------------
<PAGE>

1.   Exercise.
     --------

(a)  Right to Exercise; Exercise Price.  The Holder shall have the right to
exercise this Warrant at any time and from time to time during the period
beginning on the Issue Date and ending on the Expiration Date as to all or any
part of the shares of Common Stock covered hereby (the "Warrant Shares"). The
                                                        --------------
"Exercise Price" payable by the Holder in connection with the exercise of this
 --------------
Warrant shall be $17.536, equal to one hundred and twenty percent (120%) of the
Market Price on the Issue Date (subject to adjustment as provided herein).
"Market Price" on any day means the average Closing Bid Price for the Common
-------------
Stock during the period of five (5) consecutive Trading Days immediately
preceding (but not including) such day.

(b)  Exercise Notice.  In order to exercise this Warrant, the Holder shall send
by facsimile transmission, at any time prior to 7:00 p.m., eastern time, on the
Business Day (as defined below) on which the Holder wishes to effect such
exercise (the "Exercise Date"), to the Company a copy of the notice of exercise
               -------------
in the form attached hereto as Exhibit A (the "Exercise Notice") stating the
                                               ---------------
number of Warrant Shares as to which such exercise applies and the calculation
therefor. As used herein, "Business Day" shall mean any day on which the Nasdaq
                           ------------
National Market and commercial banks in the cities of San Francisco and New York
are open for business. The Holder shall thereafter deliver to the Company the
original Exercise Notice, the original Warrant and the Exercise Price. In the
case of a dispute as to the calculation of the Exercise Price or the number of
Warrant Shares issuable hereunder (including without limitation the calculation
of any adjustment to the Exercise Price pursuant to Section 6 below), the
Company shall promptly issue to the Holder the number of Warrant Shares that are
not disputed and shall submit the disputed calculations to the Company's
independent accountant within three (3) Business Days following the Exercise
Date. The Company shall cause such accountant to calculate the Exercise Price
and/or the number of Warrant Shares issuable hereunder and to notify the Company
and the Holder of the results in writing no later than two Business Days
following the day on which such accountant received the disputed calculations.
Such accountant's calculation shall be deemed conclusive absent manifest error.
The fees of any such accountant shall be borne by the party whose calculations
were most at variance with those of such accountant.

(c)  Cancellation of Warrant.  This Warrant shall be canceled upon its exercise
and, if this Warrant is exercised in part, the Company shall, at the time that
it delivers Warrant Shares to the Holder pursuant to such exercise as provided
herein, issue a new warrant, and deliver to the Holder a certificate
representing such new warrant, with terms identical in all respects to this
Warrant (except that such new warrant shall be exercisable into the number of
shares of Common Stock with respect to which this Warrant shall remain
unexercised); provided, however, that the Holder shall be entitled to exercise
              --------  -------
all or any portion of such new warrant at any time following the time at which
this Warrant is exercised, regardless of whether the Company has actually issued
such new warrant or delivered to the Holder a certificate therefor.

2.   Delivery of Warrant Shares Upon Exercise.  Upon receipt of a Exercise
     ----------------------------------------
Notice pursuant to paragraph 1 above, the Company shall, (A) in the case of a
Cashless Exercise (as defined below), no

                                      -2-
<PAGE>

later than the close of business on the third (3rd) Business Day following the
Exercise Date set forth in such Exercise Notice, (B) in the case of a Cash
Exercise (as defined below) no later than the close of business on the later to
occur of (i) the third (3rd) Business Day following the Exercise Date set forth
in such Exercise Notice and (ii) such later date on which the Company shall have
received payment of the Exercise Price, and (C) with respect to Warrant Shares
which are disputed as described in paragraph 1(b) above, and required to be
delivered by the Company pursuant to the accountant's calculations described
therein, the close of business on the third (3rd) Business Day following the
determination made pursuant to paragraph 1(b) (the "Delivery Date"), issue and
                                                    -------------
deliver or caused to be delivered to the Holder the number of Warrant Shares as
shall be determined as provided herein. The Company shall effect delivery of
Warrant Shares to the Holder by, as long as the Company's designated transfer
agent for the Common Stock (the "Transfer Agent") participates in the Depository
                                 --------------
Trust Company ("DTC") Fast Automated Securities Transfer program ("FAST"),
                ---                                                ----
crediting the account of the Holder or its nominee at DTC (as specified in the
applicable Exercise Notice) with the number of Warrant Shares required to be
delivered, no later than the close of business on such Delivery Date. In the
event that the Transfer Agent is not a participant in FAST, or if Warrant Shares
are not otherwise eligible for delivery through FAST, or if the Holder so
specifies in an Exercise Notice or otherwise in writing on or before the
Exercise Date, the Company shall effect delivery of Warrant Shares by delivering
to the Holder or its nominee physical certificates representing such Warrant
Shares, no later than the close of business on such Delivery Date. Warrant
Shares delivered to the Holder shall not contain any restrictive legend as long
as (i) the resale of such Warrant Shares is covered by an effective Registration
Statement (as defined in the Registration Rights Agreement) and such Holder
represents in writing to the Company that such Warrant Shares have been or are
being sold pursuant to such registration statement, (ii) such Holder represents
in writing to the Company that such Warrant Shares have been publicly sold
pursuant to Rule 144 ("Rule 144") and such Holder has delivered to the Company
                       --------
customary Rule 144 broker's and seller's representation letters, or (iii) such
Holder represents in writing to the Company that sales of such Warrant Shares
may be made pursuant to Rule 144(k) under the Securities Act of 1933, as
amended, or any successor rule or provision.

3.   Failure to Deliver Warrant Shares.
     ---------------------------------

(a)  Exercise Default.  In the event that, as a result of any action or failure
to act on the part of the Company (including without limitation a failure by the
Company to have a sufficient number of shares of Common Stock authorized and
reserved for issuance pursuant to exercise of the Warrants), the Company does
not deliver to a Holder certificates representing the number of Warrant Shares
specified in the applicable Exercise Notice on or before the Delivery Date
therefor and such failure continues for ten (10) Business Days (an "Exercise
                                                                    --------
Default"), the Company shall pay to the Holder payments ("Exercise Default
-------                                                   ----------------
Payments") in the amount of (i) (N/365) multiplied by (ii) the aggregate
--------                                -------------
Exercise Price for the Warrant Shares which are the subject of such Exercise
Default multiplied by (iii) the lower of twenty four percent (24%) and the
        -------------
maximum rate permitted by applicable law, where "N" equals the number of days
elapsed between the original Delivery Date for such Warrant Shares and the date
on which all of such Warrant Shares are issued and delivered to the Holder.
Amounts payable under this subparagraph 3(a) shall be paid to the Holder in
immediately

                                      -3-
<PAGE>

available funds on or before the fifth (5th) Business Day of the calendar month
immediately following the calendar month in which such amount has accrued.

(b)  Buy-in.  Nothing herein shall limit a Holder's right to pursue actual
damages for the Company's failure to issue and deliver Warrant Shares in
connection with an exercise on the applicable Delivery Date (including, without
limitation, damages relating to any purchase of shares of Common Stock by the
Holder to make delivery on a sale effected in anticipation of receiving Warrant
Shares upon exercise, such damages to be in an amount equal to (A) the aggregate
amount paid by the Holder for the shares of Common Stock so purchased minus (B)
                                                                      -----
the aggregate amount of net proceeds, if any, received by the Holder from
the sale of the Warrant Shares issued by the Company pursuant to such exercise),
and the Holder shall have the right to pursue all remedies available to it at
law or in equity (including, without limitation, a decree of specific
performance and/or injunctive relief).

(c)  Reduction of Exercise Price. In the event that, as a result of any action
or failure to act on the part of the Company (including without limitation a
failure by the Company to have a sufficient number of shares of Common Stock
authorized and reserved for issuance pursuant to exercise of the Warrants), a
Holder has not received certificates representing the Warrant Shares by the
tenth (10th) Business Day following an Exercise Default, the Holder may, upon
written notice to the Company, regain on such Business Day the rights of a
Holder of this Warrant, or part thereof, with respect to the Warrant Shares that
are the subject of such Exercise Default, and the Exercise Price for such
Warrant Shares shall be reduced by one percent (1%) for each day beyond such
10th Business Day in which the Exercise Default continues. In such event, the
Holder shall retain all of the Holder's rights and remedies with respect to the
Company's failure to deliver such Warrant Shares (including without limitation
the right to receive the cash payments specified in subparagraph 3(a) above). In
no event shall any adjustment pursuant to this paragraph (c) result in an
Exercise Price that is less than the Closing Bid Price for the Common Stock on
the Issue Date (as such Closing Bid Price may be adjusted for stock splits,
stock dividends and similar events).

(d)   Holder of Record.  Each Holder shall, for all purposes, be deemed to have
become the holder of record of Warrant Shares on the Exercise Date of this
Warrant, irrespective of the date of delivery of such Warrant Shares. Nothing in
this Warrant shall be construed as conferring upon the Holder hereof any rights
as a stockholder of the Company prior to the Exercise Date.

  4.   Exercise Limitations.
       --------------------

  In no event shall a Holder be permitted to exercise this Warrant, or part
thereof, with respect to Warrant Shares in excess of the number of such shares,
upon the issuance of which, (x) the number of shares of Common Stock
beneficially owned by the Holder plus (y) the number of shares of Common Stock
                                 ----
issuable upon such exercise plus (z) the number of shares of Common Stock that
                            ----
was issued to the Holder pursuant to this Warrant during the period of sixty
(60) days immediately preceding such exercise, would be equal to or exceed 9.99%
of the number of shares of Common Stock then issued and outstanding, it being
the intent of the Company and the Holder that no Holder be deemed at any

                                      -4-
<PAGE>

time to have the power to acquire, vote or dispose of greater than 9.99% of the
number of shares of Common Stock issued and outstanding during any sixty-day
period. To the extent that the limitation contained in this paragraph 4 applies,
the submission of a Conversion Notice by the Holder shall be deemed to be the
Holder's representation that this Warrant is exercisable pursuant to the terms
hereof and the Company shall be entitled to rely on such representation without
making any further inquiry. Nothing contained herein shall be deemed to restrict
the right of a Holder to exercise this Warrant, or part thereof, at such time as
such exercise will not violate the provisions of this Section 4. The limitation
contained in this Section 4 shall not apply if the original Holder hereof
elected in the Securities Purchase Agreement not to be subject to such
limitation.

5.   Payment of the Exercise Price.  The Holder may pay the Exercise Price in
     -----------------------------
either of the following forms or, at the election of Holder, a combination
thereof:

(a)  Cash Exercise: by delivery of immediately available funds.

(b)  Cashless Exercise: by surrender of this Warrant to the Company together
with a notice of cashless exercise, in which event the Company shall issue
to the Holder the number of Warrant Shares determined as follows:

                   X = Y x (A-B)/A

where:             X = the number of Warrant Shares to be issued to the Holder.

                   Y = the number of Warrant Shares with respect to which this
                   Warrant is being exercised.

                   A = the average of the Closing Bid Prices of the Common Stock
                   for the five (5) Trading Days immediately prior to (but not
                   including) the Exercise Date.

                   B = the Exercise Price;

provided, however, that the Holder may exercise this Warrant pursuant to a
--------  -------
Cashless Exercise only if, on the Exercise Date, the resale of Warrant Shares is
not covered by an effective Registration Statement (as defined in the
Registration Rights Agreement) that is available to the Holder on such date.

For purposes of Rule 144 under the Securities Act of 1933, as amended, it is
intended, understood and acknowledged that the Warrant Shares issued in a
cashless exercise transaction shall be deemed to have been acquired by the
Holder, and the holding period for the Warrant Shares shall be deemed to have
been commenced, on the Issue Date to the extent permitted by law.

6.  Anti-Dilution Adjustments; Distributions; Other Events. The Exercise Price
    ------------------------------------------------------
and the

                                      -5-
<PAGE>

number of Warrant Shares issuable hereunder shall be subject to adjustment from
time to time as provided in this Section 6. In the event that any adjustment of
the Exercise Price or number of Warrant Shares as required herein results in a
fraction of a cent or fraction of a share, as applicable, such exercise Price or
number of Warrant Shares shall be rounded upon or down to the nearest cent or
share, as applicable.

(a)  Adjustment of Exercise Price upon Issuance of Common Stock below the
     --------------------------------------------------------------------
Exercise Price. In the event that the Corporation issues Common Stock,
--------------
whether upon the exercise of rights, warrants, securities convertible or
exercisable into Common Stock or otherwise (including under the Equity Line), at
a price per share that is lower than the Exercise Price in effect on any date
following the Issue Date (a "Dilutive Issuance"), the Exercise Price shall be
                             -----------------
reduced, on a weighted average basis, to such lower price. Any adjustment made
herein on a "weighted average basis" as a result of a Dilutive Issuance of
Common Stock shall be calculated taking into account both (x) the difference
between the price at which such Common Stock is issued and the applicable
Exercise Price and (y) the number of shares of Common Stock being issued
relative to the number of shares of Common Stock into which this Warrant is
exercisable at the Exercise Price on the date of such issuance. By way of
example, assuming that (a) the Exercise Price is $9.00, (b) this Warrant is
exercisable at the Exercise Price into 1.04 million shares of Common Stock, (c)
the amount of the Dilutive Issuance is 520,00 shares, and (d) the purchase price
per share of Common Stock being issued in the Dilutive Issuance is $5, the Fixed
Conversion Price as adjusted would be equal to $7.00. In no event shall any
adjustment pursuant hereto result in (i) an Exercise Price that exceeds the
Exercise Price that would otherwise apply in the absence of such adjustment or
(ii) an Exercise Price that is less than the Closing Bid Price for the Common
Stock on the Issue Date (as such Closing Bid Price may be adjusted for stock
splits, stock dividends and similar events).

(b)  Exceptions to Adjustment of Exercise Price.  No adjustment to the Exercise
     ------------------------------------------
Price will be made as a result of the issuance of any securities of the Company
issued pursuant to (i) an employee stock option, purchase or benefit plan or
program duly adopted by the Company; (ii) any options, warrant, convertible
securities or rights or agreements to purchase securities of the Company
outstanding on the date hereof; (iii) any firm-commitment underwritten public
offerings of securities in an amount not less than $40,000,000; (iv) any
securities issued for consideration other than cash pursuant to a merger,
consolidation, acquisition or similar business combination with a third party,
the primary purpose of which is not the raising of capital (provided that, in
any such event, the provisions of paragraphs 6(c), 6(d), 6(e), 6(f) and 6(g)
will remain in effect); (v) shares of Common Stock issued upon conversion of the
Preferred Shares or exercise of the Warrants; (vi) any debt or other securities
issued pursuant to any equipment leasing arrangement or debt financing from a
bank or other financial institution, or (vii) any securities issued in
connection with strategic transactions involving the Company and third parties,
including by way of example (A) joint ventures, manufacturing, marketing or
distribution arrangements or (B) technology transfer or development
arrangements; provided, that, in any such event, the provisions of paragraphs
6(c), 6(d), 6(e), 6(f) and 6(g) will remain in effect.

(c)  Subdivision or Combination of Common Stock.  If the Company, at any time
     ------------------------------------------
after

                                      -6-
<PAGE>

the initial issuance of this Warrant, subdivides (by any stock split, stock
dividend, recapitalization, reorganization, reclassification or otherwise) its
shares of Common Stock into a greater number of shares, then after the date of
record for effecting such subdivision, the Exercise Price in effect immediately
prior to such subdivision will be proportionately reduced. If the Company, at
any time after the initial issuance of this Warrant, combines (by reverse stock
split, recapitalization, reorganization, reclassification or otherwise) its
shares of Common Stock into a smaller number of shares, then, after the date of
record for effecting such combination, the Exercise Price in effect immediately
prior to such combination will be proportionally increased

(d)  Distributions.  If the Company or any of its subsidiaries shall at any time
     -------------
distribute to holders of Common Stock (or to a holder, other than the Company,
of the common stock of any such subsidiary) cash, evidences of indebtedness or
other securities or assets (other than cash dividends or distributions payable
out of earned surplus or net profits for the current or the immediately
preceding year) including any dividend or distribution in shares of capital
stock of a subsidiary of the Company (collectively, a "Distribution") then, in
                                                       ------------
any such case, the Holder of this Warrant shall be entitled to receive, at the
same time as such assets are received by a holder of such stock, at the option
of such Holder, either (A) an amount and type of such Distribution as though
such Holder were a holder on the record date therefor of a number of shares of
Common Stock into which this Warrant is exercisable as of such record date (such
number of shares to be determined at the Exercise Price then in effect and
without regard to any limitation on exercise of this Warrant that may exist
pursuant to the terms hereof or otherwise) , or (B) a reduction in the Exercise
Price for the Preferred Shares held by such Holder as of the record date for
such Distribution, such reduction to be effected by reducing the Exercise Price
in effect on the Business Day immediately preceding such record date by an
amount equal to the fair market value of the assets so distributed divided by
the number of shares of Common Stock to which such Distribution is made, such
fair market value to be reasonably determined by an investment banking firm
selected by the Holder and which is reasonably acceptable to the Company.

(e)  Consolidation or Merger.  In the event of a merger, consolidation, business
     -----------------------
combination, tender offer, exchange of shares, recapitalization, reorganization,
redemption or other similar event, as a result of which shares of Common Stock
of the Company shall be changed into the same or a different number of shares of
the same or another class or classes of stock or securities or other assets of
the Company or another entity or there is a sale of all or substantially all the
Company's assets (a "Corporate Change"), then this Warrant shall be exercisable
                     ----------------
into such class and type of securities or other assets as the Holder would have
received had the Holder exercised this Warrant immediately prior to such
Corporate Change; provided, however, that Company may not effect any Corporate
Change unless (i) it (or, in the case of a tender offer, the offering party)
first shall have given ten (10) days' notice to the Holder hereof of any
Corporate Change and makes a public announcement of such event before or at the
same time that it gives such notice (it being understood that the filing by the
Company of a Form 8-K for the purpose of disclosing the anticipated consummation
of the Corporate Change shall constitute such a notice for purposes of this
provision) and (ii) it requires the resulting successor or acquiring entity (if
not the Company) to assume by written instrument the obligations of the Company
hereunder and under the Securities Purchase Agreement and the

                                      -7-
<PAGE>

Registration Rights Agreement.

(f) Adjustments; Additional Shares, Securities or Assets.  In the event that at
    ----------------------------------------------------
any time, as a result of an adjustment made pursuant to this paragraph 6, the
Holder of this Warrant shall, upon exercise of this Warrant, become entitled to
receive securities or assets (other than Common Stock) then, wherever
appropriate, all references herein to shares of Common Stock shall be deemed to
refer to and include such shares and/or other securities or assets; and
thereafter the number of such shares and/or other securities or assets shall be
subject to adjustment from time to time in a manner and upon terms as nearly
equivalent as practicable to the provisions of this paragraph 6. Any adjustment
made herein that results in a decrease in the Exercise Price shall also effect a
proportional increase in the number of shares of Common Stock into which this
Warrant is exercisable.

7.  Fractional Interests.
    --------------------

    No fractional shares or scrip representing fractional shares shall be
issuable upon the exercise of this Warrant, but on exercise of this Warrant, the
Holder hereof may purchase only a whole number of shares of Common Stock. If, on
exercise of this Warrant, the Holder hereof would be entitled to a fractional
share of Common Stock or a right to acquire a fractional share of Common Stock,
such fractional share shall be disregarded and the number of shares of Common
Stock issuable upon exercise shall be rounded up or down to the nearest whole
number of shares of Common Stock.

8.  Transfer of this Warrant.  The Holder may sell, transfer, assign, pledge
    ------------------------
or otherwise dispose of this Warrant, in whole or in part, as long as such sale
or other disposition is made pursuant to an effective registration statement or
an exemption to the registration requirements of the Securities Act of 1933, as
amended, and applicable state laws.  Upon such transfer or other disposition,
the Holder shall deliver a written notice to Company, substantially in the form
of the Transfer Notice attached hereto as Exhibit B (the "Transfer Notice"),
                                                          ---------------
indicating the person or persons to whom this Warrant shall be transferred and,
if less than all of this Warrant is transferred or this Warrant is transferred
in parts, the number of Warrant Shares to be covered by the part of this Warrant
to be transferred to each such person. Within three (3) Business Days of
receiving a Transfer Notice and the original of this Warrant, the Company  shall
deliver to the each transferee designated by the Holder a Warrant or Warrants of
like tenor and terms for the appropriate number of Warrant Shares.
Notwithstanding the foregoing, no Holder may knowingly and voluntarily sell this
Warrant (or any portion thereof) to an entity that is a competitor of the
Company.

9.  Benefits of this Warrant.
    ------------------------

    Nothing in this Warrant shall be construed to confer upon any person other
than the Holder of this Warrant any legal or equitable right, remedy or claim
under this Warrant and this Warrant shall be for the sole and exclusive benefit
of the Holder of this Warrant.

10. Loss, theft, destruction or mutilation  of Warrant.
    --------------------------------------------------

                                      -8-
<PAGE>

    Upon receipt by the Company of evidence 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 the Company, and upon surrender
of this Warrant, if mutilated, the Company shall execute and deliver a new
Warrant of like tenor and date.

11.  Notice or Demands.
     -----------------

     Except as otherwise provided herein, any notice, demand or request required
or permitted to be given pursuant to the terms of this Warrant shall be in
writing and shall be deemed given (i) when delivered personally or by verifiable
facsimile transmission (with an original to follow) on or before 5:00 p.m.,
eastern time, on a Business Day or, if such day is not a Business Day, on the
next succeeding Business Day, (ii) on the next Business Day after timely
delivery to a nationally-recognized overnight courier and (iii) on the Business
Day actually received if deposited in the U.S. mail (certified or registered
mail, return receipt requested, postage prepaid), addressed as follows:

     If to the Company:

     Pilot Network Services, Inc.
     1080 Marina Village Parkway
     Alameda, California 94501
     Attn: Chief Financial Officer
     Tel: (510) 433-7800
     Fax: (510)

     with a copy to:

     Heller Ehrman White & McAuliffe LLP
     333 Bush Street
     San Francisco, California  94104
     Attn:  Timothy G. Hoxie
     Tel:  (415) 772-6000
     Fax:  (415) 772-6268

and if to the Holder, to such address as shall be designated by the Holder in
writing to the Company.

12.  Applicable Law.
     --------------

     This Warrant is issued under and shall for all purposes be governed by and
construed in accordance with the laws of the state of Delaware, without giving
effect to conflict of law provisions thereof.

                                      -9-
<PAGE>

                 [REMAINDER OF PAGE INTENTIONALLY LEFT BLANK]

                                      -10-
<PAGE>

  IN WITNESS WHEREOF, the undersigned has executed this Warrant as of the 28th
day of June, 2000.


                         PILOT NETWORK SERVICES, INC.


                         By:  /s/ M. Marketta Silvera
                              _____________________________________________
                              Name: M. Marketta Silvera
                              Title: President and Chief Executive Officer

                                      -11-
<PAGE>

                                                            EXHIBIT A to WARRANT
                                                            --------------------

                                EXERCISE NOTICE
                                ---------------


  The undersigned Holder hereby irrevocably exercises the right to purchase
________________ of the shares of Common Stock ("Warrant Shares") of PILOT
                                                 --------------
NETWORK SERVICES, INC. evidenced by the attached Warrant (the "Warrant").
                                                               -------
Capitalized terms used herein and not otherwise defined shall have the
respective meanings set forth in the Warrant. Unless otherwise specified in
writing to the Company, the undersigned represents to the Company that the
shares of Common Stock covered by this notice have been or will be sold pursuant
to the terms of an effective registration statement.

1.   Form of Exercise Price.  The Holder intends that payment of the Exercise
Price shall be made as:

     ______ a Cash Exercise with respect to _________________ Warrant Shares;
               -------------
      and/or

     ______ a Cashless Exercise with respect to _________________ Warrant
              -----------------
              Shares.


2.   Payment of Exercise Price.  In the event that the Holder has elected a
Cash Exercise with respect to some or all of the Warrant Shares to be issued
pursuant hereto, the Holder shall pay the sum of $________________ to the
Company in accordance with the terms of the Warrant.

3.   Delivery of Warrant Shares.  The Company shall deliver to the Holder
_____________ Warrant Shares in accordance with the terms of the Warrant.



Date: ______________________


___________________________________
  Name of Registered Holder

By:  _______________________________
     Name:
     Title:

                                      -12-
<PAGE>

                                                            EXHIBIT B to WARRANT
                                                            --------------------

                                TRANSFER NOTICE
                                ---------------


FOR VALUE RECEIVED, the undersigned Holder of the attached Warrant hereby sells,
assigns and transfers unto the person or persons named below the right to
purchase ___________shares of the Common Stock of PILOT NETWORK SERVICES, INC.
evidenced by the attached Warrant.


Date: ______________________


___________________________________
  Name of Registered Holder

By:  _______________________________
    Name:
    Title:

Transferee Name and Address:

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

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

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





                                      -13-