<PAGE>

                                                                    EXHIBIT 10.3

     THIS WARRANT AND THE SHARES ISSUABLE HEREUNDER HAVE NOT BEEN REGISTERED
UNDER THE SECURITIES ACT OF 1933, AS AMENDED, AND MAY NOT BE SOLD, PLEDGED, OR
OTHERWISE TRANSFERRED WITHOUT AN EFFECTIVE REGISTRATION THEREOF UNDER SUCH ACT
OR PURSUANT TO RULE 144 OR AN OPINION OF COUNSEL, WHICH MAY BE IN-HOUSE COUNSEL,
IN A FORM REASONABLY SATISFACTORY TO THE COMPANY THAT SUCH REGISTRATION IS NOT
REQUIRED.

                      ___________________________________

                           WARRANT TO PURCHASE STOCK

Warrant to Purchase 200,000 Shares    Issue Date:              December 28, 1999
of the Common Stock of Pilot          Expiration Date:         December 28, 2002
Network Services, Inc.                Initial Exercise Price:  $25.00 per share


     FOR VALUE RECEIVED, PILOT NETWORK SERVICES, INC., a Delaware corporation
(the "Company"), hereby certifies that PRIMUS TELECOMMUNICATIONS GROUP,
      -------
INCORPORATED ("Holder") is entitled to purchase the number of fully paid and
               ------
non-assessable shares of common stock (the "Shares") of the Company at the
                                            ------
initial exercise price per Share (the "Warrant Price") all as set forth above
                                       -------------
and as adjusted pursuant to Article 2 of this Warrant, subject to the provisions
and upon the terms and conditions set forth in this Warrant.

ARTICLE 1.  EXERCISE.

     1.1  Method of Exercise.  This Warrant is exercisable, in whole or in part,
at any time and from time to time on or before the Expiration Date set forth
above; provided, however, that this Warrant may not be exercised until all
applicable requirements, if any, of the Hart-Scott-Rodino Antitrust Improvements
Act of 1976, as amended (the "HSR Act"), have been satisfied.  The Company
                              -------
agrees to file and cooperate with Holder in filing all documents required to
comply with the HSR Act.  Holder may exercise this Warrant by delivering a duly
executed Notice of Exercise in substantially the form attached as Appendix 1 to
the principal office of the Company.  Holder shall also deliver to the Company a
wire transfer or certified check for the aggregate Warrant Price for the Shares
being purchased.

     1.2  Delivery of Certificate and New Warrant. Promptly after Holder
exercises this Warrant, the Company shall deliver at its expense to Holder
certificates for the Shares acquired and, if this Warrant has not been fully
exercised and has not expired, a new Warrant representing the Shares not so
acquired.

     1.3  Replacement of Warrants. On receipt of evidence reasonably
satisfactory to the Company of the loss, theft, destruction or mutilation of
this Warrant and, in the case of loss, theft or destruction, on delivery of an
indemnity agreement reasonably satisfactory in form and amount to the Company
or, in the case of mutilation, on surrender and cancellation of this
<PAGE>

Warrant, the Company shall execute and deliver, in lieu of this Warrant, a new
warrant of like tenor.

     1.4  Fractional Shares.  No fractional Shares shall be issuable upon
exercise of the Warrant and the number of Shares to be issued shall be rounded
down to the nearest whole Share.  If a fractional share interest arises upon any
exercise of the Warrant, the Company shall eliminate such fractional share
interest by paying Holder an amount computed by multiplying the fractional
interest by the fair market value of a full Share.

ARTICLE 2.  ADJUSTMENTS TO THE SHARES.

     2.1  Stock Dividends, Splits, Etc.  If the Company declares or pays a
dividend on its common stock payable in common stock or other securities,
combines its outstanding shares of common stock, makes a distribution of capital
stock to holders of its common stock or subdivides the outstanding common stock
into a greater amount of common stock, then the number of shares for which this
Warrant may be exercised shall be proportionately adjusted such that, upon
exercise of this Warrant, Holder shall receive a proportionate number of
securities to which Holder would have been entitled had Holder owned the Shares
of record as of the date the applicable event occurred, and the per share
Warrant Price shall be adjusted so that the aggregate Warrant Price remains the
same.

     2.2  Reclassification.  Upon any reclassification that results in a change
of the number and/or class of the securities issuable upon exercise of this
Warrant, Holder shall be entitled to receive, upon exercise of this Warrant, the
number and kind of securities and property that Holder would have received for
the Shares if this Warrant had been exercised immediately before such
reclassification. The Company shall promptly issue to Holder a new Warrant for
such new securities or other property. The new Warrant shall provide for
adjustments which shall be as nearly equivalent as may be practicable to the
adjustments provided for in this Article 2 including, without limitation,
adjustments to the Warrant Price and the number of securities or property
issuable upon exercise of the new Warrant. The provisions of this Section 2.2
shall similarly apply to successive reclassifications.

     2.3  Adjustments for Business Combinations, Etc.  Upon 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 (a "Consolidation Event"),
                                                        -------------------
then the Holder shall be entitled to receive upon such transfer, merger or
consolidation becoming effective, and upon payment of the Warrant Price, the
number of shares or other securities or property of the Company or of the
successor corporation resulting from such merger or consolidation, which would
have been received by the Holder for the shares of stock subject to this Warrant
had this Warrant been exercised immediately prior to such transfer, merger or
consolidation becoming effective or to the applicable record date thereof, as
the case may be. The Company shall not effect any Consolidation Event unless the
resulting successor or acquiring entity (if not the Company) assumes by written
instrument the obligation to deliver to the Holder such shares of stock and/or
other securities as the Holder is entitled to receive had this Warrant been
exercised in accordance with the foregoing. Such successor shall promptly issue
to Holder a new Warrant for such new securities or other property. The new
Warrant shall provide for adjustments which shall be as nearly equivalent as may
be practicable to the adjustments provided for in this Article 2

                                       2
<PAGE>

including, without limitation, adjustments to the Warrant Price and the number
of securities or property issuable upon exercise of the new Warrant. The
provisions of this Section 2.3 shall similarly apply to successive
reclassifications, exchanges, substitutions, or other events.

     2.4  No Impairment.  The Company shall not, by amendment of its Certificate
of Incorporation or through a 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 to be
observed or performed under this Warrant by the Company, but shall at all times
in good faith assist in carrying out of all the provisions of this Article 2 and
in taking all such action as may be necessary or appropriate to protect Holder's
rights under this Article against impairment.

     2.5  Certificate as to Adjustments.  Upon each adjustment of the Warrant
Price, the Company shall promptly compute such adjustment and furnish Holder
with a certificate of its Chief Financial Officer setting forth such adjustment
and the facts upon which such adjustment is based.  The Company shall, upon
written request, furnish Holder a certificate setting forth the Warrant Price in
effect upon the date thereof and the series of adjustments leading to such
Warrant Price.

ARTICLE 3.  REPRESENTATIONS AND COVENANTS OF THE COMPANY.

     3.1  Validly Issued Shares.  The Company hereby represents and warrants to
the Holder that the Shares that may be issued upon the exercise of the purchase
right represented by this Warrant shall, upon issuance, be duly authorized,
validly issued, fully paid and non-assessable, and free of any liens and
encumbrances except for restrictions on transfer provided for herein or under
applicable federal and state securities laws.

     3.2  Valid Authorization.  The Company hereby represents and warrants to
the Holder that the execution and delivery of this Warrant by the Company have
been duly authorized by all necessary corporate action on the part of the
Company and no additional consent or approval by any third party or governmental
authority is required for the execution and delivery by the Company of this
Warrant, other than those required under the HSR Act under certain
circumstances.

     3.3  Notice of Certain Events.  If the Company proposes at any time (a) to
declare any dividend or distribution upon its common stock, whether in cash,
property, stock, or other securities (other than a regular cash dividend); (b)
to offer for subscription pro rata to the holders of common stock any additional
shares of stock of any class or series or other rights; (c) to effect any
reclassification or recapitalization of common stock; or (d) to merge or
consolidate with or into any other corporation, or sell or convey all or
substantially all of its assets, or to liquidate, dissolve or wind up; then, in
connection with each such event, the Company shall give Holder (1) at least 10
business days prior written notice of the date on which a record will be taken
for such dividend, distribution, or subscription rights (and specifying the date
on which the holders of common stock will be entitled thereto) or for
determining rights to vote, if any, in respect of the matters referred to in (c)
and (d) above; and (2) in the case of the matters referred to in (c) and (d)
above, the same notice that is provided to stockholders of the Company including
when the same will take place (and specifying the date on which the holders of
common stock will be

                                       3
<PAGE>

entitled to exchange their common stock for securities or other property
deliverable upon the occurrence of such event).

     3.4  Reservation of Shares.  The Company shall at all times reserve for
issuance pursuant to this Warrant such number of shares of Common Stock as are
issuable upon exercise of this Warrant.

     3.5  Periodic Reports.  The Company shall continue to file with the
Securities and Exchange Commission all periodic reports required under the
Securities Exchange Act of 1934 during the term of this Warrant; provided,
however, that such obligation may terminate in the event of a merger or
consolidation of the Company in which the Company is not the surviving entity,
the acquisition of substantially all of the voting stock of the Company by a
single entity or group of related entities or a sale of all or substantially all
of the assets of the Company.

     3.6  NASDAQ Listing.  (a) The Company will cause the shares of Common Stock
issuable upon exercise of this Warrant to be listed for trading on the NASDAQ
National Market System.

          (b)  The Company will use its best efforts to cause its common stock
to remain listed for trading on the NASDAQ National Market System or the New
York Stock Exchange; provided, however, that such obligation may terminate in
the event of or merger or consolidation of the Company in which the Company is
not the surviving entity, the acquisition of substantially all of the voting
stock of the Company by a single entity or group of related entities or a sale
of all or substantially all of the assets of the Company.

     3.7  Registration Rights.  The Company agrees to use its best efforts to
obtain by January 31, 2000 the requisite consent of the holders of registrable
stock under that certain Amended and Restated Investors' Rights Agreement, dated
as of March 31, 1997, among the Company and the investors listed on Schedule I
thereto, as amended by that certain Amendment to Investors' Rights Agreement
dated as of December 22, 1997 and that certain Second Amendment to Investors'
Rights Agreement dated February 26, 1998, including the obligation of the
Company to amend such agreement to grant registration rights to Greyrock Capital
(the "Existing Investors' Rights Agreement"), necessary to amend such agreement
      ------------------------------------
(a) to grant to the Holder demand registration rights with respect to the Shares
beginning six months after the date hereof and (b) to grant piggyback
registration rights to the Holder with respect to the Shares on the same terms
and conditions as are generally applicable to the other holders of registrable
stock thereunder; provided, however, that the Holder will have the right to
demand a registration if the Holder proposes to register all of the Shares. If
the Company is unable to so amend the Existing Investors' Rights Agreement, the
Company shall enter into a registration rights agreement with the Holder in a
form reasonably acceptable to the Holder.

ARTICLE 4.  MISCELLANEOUS.

     4.1  This Warrant and the Shares shall be imprinted with a legend in
substantially the following form:

     THIS SECURITY HAS NOT BEEN REGISTERED UNDER THE SECURITIES ACT OF 1933, AS
AMENDED, AND MAY NOT BE SOLD, PLEDGED OR OTHERWISE

                                       4
<PAGE>

TRANSFERRED WITHOUT AN EFFECTIVE REGISTRATION THEREOF UNDER SUCH ACT OR PURSUANT
TO RULE 144 OR AN OPINION OF COUNSEL, WHICH MAY BE IN-HOUSE COUNSEL, IN A FORM
REASONABLY SATISFACTORY TO THE COMPANY THAT SUCH REGISTRATION IS NOT REQUIRED.

     4.2  Compliance with Securities Laws on Transfer.  This Warrant and the
Shares issuable upon exercise of this Warrant may not be transferred or assigned
in whole or in part without compliance with applicable federal and state
securities laws (including, without limitation, the delivery of investment
representation letters and legal opinions satisfactory to the Company, if
requested by the Company). The Company shall not require Holder to provide an
opinion of counsel if the transfer is to an affiliate of Holder or if there is
no material question as to the availability of current information as referenced
in Rule 144(c), Holder represents that it has complied with Rule 144(d) and (e),
the selling broker represents that it has complied with Rule 144(f), and the
Company is provided with a copy of Holder's notice of proposed sale.

     4.3  Transfer Procedure.  Subject to the provisions of Sections 4.1 and
4.2, Holder may transfer all or part of this Warrant by giving the Company
notice of the portion of the Warrant being transferred setting forth the name,
address and taxpayer identification number of the transferee and surrendering
this Warrant to the Company for reissuance to the transferee (and Holder if
applicable).

     4.4  Notices.  Any notice required or permitted by this Warrant shall be in
writing and shall be deemed sufficient (i) upon delivery, when delivered
personally or sent by facsimile, (ii) on the next business day after dispatch
when delivered by overnight courier or sent by telegram, or (iii) on the fifth
business day after being deposited in the U.S. mail, as certified or registered
mail, with postage prepaid, addressed (a) if to the Company, to 1080 Marina
Village Parkway, Alameda, California  94501, Attn:  Chief Financial Officer,
facsimile number (510) 433-7811, with a copy to Orrick, Herrington & Sutcliffe
LLP, 400 Sansome Street, San Francisco, California  94111, Attn:  Peter
Lillevand, Esq., facsimile number (415) 773-5759 or (b) if to the Holder, to
1700 Old Meadow Road, Third Floor, McLean, Virginia  22102, Attn:  David P.
Slotkin, Esq., Deputy General Counsel, with a copy to Hogan & Hartson L.L.P.,
Columbia Square, 555 Thirteenth Street, N.W., Washington, D.C. 20004, Attn:
David W. Bonser, Esq., or such other addresses as either party may from time to
time specify in writing.

     4.5  Waiver.  This Warrant and any term hereof may be changed, waived,
discharged or terminated only by an instrument in writing signed by the party
against which enforcement of such change, waiver, discharge or termination is
sought.

     4.6  Governing Law.  This Warrant shall be governed by and construed in
accordance with the laws of the State of Delaware, without giving effect to its
principles regarding conflicts of law.

                                    PILOT NETWORK SERVICES, INC.

                                    By  /s/ William C. Leetham
                                        --------------------------------------
                                        Name:  William C. Leetham
                                        Title: Chief Financial Officer

                                       5
<PAGE>

                                  APPENDIX 1

                              NOTICE OF EXERCISE
                              ------------------

     1.   The undersigned hereby elects to purchase _________________ shares of
the Common Stock of Pilot Network Services, Inc. pursuant to the terms of the
attached Warrant, and tenders herewith payment of the purchase price of such
shares in full.

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



                    ___________________________
                                   (NAME)


                    ___________________________
                    ___________________________
                    ___________________________
                                (ADDRESS)

     3.   The undersigned represents it is acquiring the shares solely for its
own account and not as a nominee for any other party and not with a view toward
the resale or distribution thereof except in compliance with applicable
securities laws.


     ___________________________
     (Signature)



     ___________________________
     (Date)

                                       6