<PAGE>

                                                                  Exhibit 4.06

                      AMENDMENT NO. 2 TO RIGHTS AGREEMENT


     AMENDMENT NO. 2, dated as of January 9, 2001 (this "Amendment No. 2"), to
                                                         ---------------
the Rights Agreement, dated as of January 27, 1999 (the "Rights Agreement"),
                                                         ----------------
between Exodus Communications, Inc., a Delaware corporation (the "Company") and
                                                                  -------
Fleet National Bank. (the "Rights Agent").
                           ------------

     WHEREAS, the Company and the Rights Agent have heretofore executed and
entered into the Rights Agreement;

     WHEREAS, Section 27 of the Rights Agreement provides that the Company may,
and the Rights Agent shall, if the Company directs, supplement or amend the
Rights Agreement without the approval of any holders of Right Certificates to
make any provisions or changes with respect to the Rights which the Company may
deem necessary or desirable;

     WHEREAS, pursuant to Section 27 of the Rights Agreement, the Board of
Directors of the Company has determined that an amendment to the Rights
Agreement is necessary and desirable to effect the following and the Company and
the Rights Agent desire to evidence such amendment in writing;

     WHEREAS, the Company, Exodus Acquisition Corp., a Delaware Corporation and
a direct wholly-owned subsidiary of the Company ("Exodus Merger Sub"), Global
                                                  -----------------
Crossing North America, Inc., a New York corporation ("Global Crossing NA"),
                                                       ------------------
Global Crossing GlobalCenter Holdings, Inc., a Delaware corporation and a direct
wholly-owned subsidiary of Global Crossing NA ("GCG"), GlobalCenter Holding Co.,
                                                ---
a Delaware corporation and an indirect wholly-owned subsidiary of Global
Crossing NA ("GlobalCenter Holding"), and GlobalCenter Inc., a Delaware
              --------------------
corporation and a direct wholly-owned subsidiary of GlobalCenter Holding and an
indirect wholly-owned subsidiary of Global Crossing NA ("GlobalCenter") have
                                                         ------------
entered into an Agreement and Plan of Merger dated as of September 28, 2000 (the
"Merger Agreement"), which provides for the merger of Exodus Merger Sub with and
 ----------------
into GlobalCenter Holding (the "Merger"), and the Board of Directors of the
                                ------
Company has approved the Merger Agreement and the Merger;

     WHEREAS, GCG is an indirect wholly-owned subsidiary of Global Crossing
Ltd., a Bermuda company, and, upon the closing of the Merger, Global Crossing
Ltd. and its Affiliates and Associates (collectively, "Global Crossing") may own
                                                       ---------------
in excess of 15% of the Company's outstanding Common Shares, and the Company
desires to exclude from the definition of an "Acquiring Person" such
shareholders, provided that their ownership does not exceed a designated
percentage of the Company's outstanding Common Shares;

     WHEREAS, the definition of an "Acquiring Person" was previously amended by
Amendment No. 1, dated as of October 20, 1999, to the Rights Agreement, which
amendment is superseded by this Amendment No. 2;
<PAGE>

     THEREFORE, in consideration of the foregoing and the mutual agreements set
forth herein, the parties hereto agree as follows:

     1.    Section 1(a) of the Rights Agreement is hereby amended and restated
           as follows:

           (a)  "Acquiring Person" shall mean any Person who or which, together
                 ----------------
                with all Affiliates and Associates of such Person, shall be the
                Beneficial Owner of 15% (the "Designated Percentage") or more of
                                              ---------------------
                the Common Shares of the Company then outstanding, but shall
                not include (i) the Company, (ii) any Subsidiary of the
                Company, (iii) any employee benefit plan of the Company or any
                Subsidiary of the Company or (iv) any entity holding Common
                Shares for or pursuant to the terms of any such plan;
                provided, however, that
                --------  -------

           (x)  in the case of in the case of FMR Corp. and its
                Affiliates and Associates, including any accounts for which FMR
                Corp. and its Affiliates and Associates have investment
                management or advisory responsibilities (including the Fidelity
                Investment mutual funds, but not including any of its
                Affiliates identified in clauses (i), (ii), (iii) or (iv) of
                this sentence), the Designated Percentage shall be 20% (the
                "Designated FMR Percentage"); and
                 -------------------------

           (y)  in the case of Global Crossing and its Affiliates and
                Associates, the Designated Percentage shall be the percentage of
                outstanding Common Shares beneficially owned by Global Crossing
                and its Affiliates and Associates immediately following the
                closing of the Merger plus one percent (1%) (the "Designated
                                                                  ----------
                Global Crossing Percentage"), provided that if Global Crossing
                --------------------------    --------
                or any of its Affiliates or Associates transfers any Common
                Shares such that the percentage of outstanding Common Shares
                beneficially owned by Global Crossing and its Affiliates and
                Associates plus one percent (1%) (the "Lower Percentage") is
                                                       ----------------
                less than the Designated Global Crossing Percentage then in
                effect, then the Designated Global Crossing Percentage shall
                be revised downward to equal the Lower Percentage (but shall
                not be reduced below the Designated Percentage).

                Notwithstanding the foregoing,

                (A)  No Person shall become an Acquiring Person if the Board
                     of Directors of the Company determines in good faith that
                     a Person who would otherwise be an Acquiring Person has
                     become such inadvertently, and such Person as promptly as
                     practicable takes such actions as may be necessary so
                     that such Person would no longer be considered an
                     Acquiring Person;

                (B)  Neither Global Crossing nor any Person which acquires
                     control of Global Crossing Ltd. shall become an Acquiring
                     Person by virtue of the acquisition of control of Global
                     Crossing Ltd. by such a Person which, together with all
                     Affiliates and Associates of such
<PAGE>

                     Person, is, immediately prior to such acquisition, the
                     Beneficial Owner of Common Shares which, when added to
                     the Common Shares of which Global Crossing is the
                     Beneficial Owner immediately prior to such acquisition,
                     would equal or exceed the Designated Global Crossing
                     Percentage, provided that such Person as promptly as
                     practicable after such acquisition takes such actions as
                     may be necessary to reduce the number of Common Shares of
                     which such Person, together with all Affiliates and
                     Associates of such Person, is the Beneficial Owner to
                     less than the Designated Global Crossing Percentage;

                (C)  No Person shall become an "Acquiring Person" as the result
                     of an acquisition of Common Shares by the Company which, by
                     reducing the number of shares outstanding, increases the
                     proportionate number of shares beneficially owned by such
                     Person and such Person's Affiliates and Associates to the
                     Designated Percentage or more of the Common Shares of the
                     Company then outstanding; provided, however, that if a
                                               --------  -------
                     Person, together with such Person's Affiliates and
                     Associates, shall become the Beneficial Owner of the
                     Designated Percentage or more of the Common Shares of the
                     Company then outstanding by reason of share purchases by
                     the Company and such Person, together with its Affiliates
                     and Associates, shall, after public announcement of such
                     share purchases by the Company, become the Beneficial
                     Owner of any additional Common Shares of the Company and
                     upon such acquisition, such Person, together with the
                     Affiliates and Associates of such Person, becomes the
                     Beneficial Owner of more than the applicable Designated
                     Percentage, then such Person shall be deemed to be an
                     "Acquiring Person."

     2.    This Amendment No. 2 to the Rights Agreement shall be governed by
           and construed in accordance with the laws of the State of Delaware
           and for all purposes shall be governed by and construed in
           accordance with the laws of such State applicable to contracts to
           be made and performed entirely within such State.

     3.    This Amendment No. 2 to the Rights Agreement may be executed in one
           or more counterparts, each of which shall be an original but all of
           which counterparts shall together constitute one and the same
           instrument. Terms not defined herein shall, unless the context
           otherwise requires, have the meanings assigned to such terms in the
           Rights Agreement.

     4.    In all respects not inconsistent with the terms and provisions of
           this Amendment No. 2 to the Rights Agreement, the Rights Agreement
           is hereby ratified, adopted, approved and confirmed. In executing
           and delivering this Amendment, the Rights Agent shall be entitled
           to all the privileges and immunities afforded to the Rights Agent
           under the terms and conditions of the Rights Agreement.
<PAGE>

     5.    If any term, provision, covenant or restriction of this Amendment
           No. 2 to the Rights Agreement is held by a court of competent
           jurisdiction or other authority to be invalid, void or
           unenforceable, the remainder of the terms, provisions, covenants
           and restrictions of this Amendment No. 2 to the Rights Agreement,
           and of the Rights Agreement, shall remain in full force and effect
           and shall in no way be effected, impaired or invalidated.

     6.    This Amendment No. 2 to the Rights Agreement shall not become
           effective until immediately prior to the time that Global Crossing
           becomes the Beneficial Owner of the Common Shares to be issued to
           GCG pursuant to the Merger Agreement, at which time this Amendment
           shall take full force and effect. If the Merger does not occur
           prior to the termination of the Merger Agreement in accordance with
           its terms, then this Amendment No. 2 shall terminate and be of no
           force or effect.

     IN WITNESS WHEREOF, the parties hereto have caused this Amendment No. 2 to
be duly executed and attested, all as of the date and year first above written.

EXODUS COMMUNICATIONS, INC.           FLEET NATIONAL BANK

By: /s/ Adam W. Wegner                By: /s/ Joshua McGinn
   -----------------------------         ------------------------------
Name:   Adam W. Wegner                Name:   Joshua McGinn
     ---------------------------           ----------------------------
Title:  Senior Vice President,        Title: Senior Account Manager
        Legal and Corporate                 ---------------------------
        Affairs, General Counsel
        and Secretary
        ------------------------