<PAGE>
                                                                   Exhibit 4.10


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                            COX CLASSIC CABLE, INC.,


                                  COXCOM, INC.

                                      and

                              JPMorgan Chase Bank

          (as successor to CHASE BANK OF TEXAS, NATIONAL ASSOCIATION),

                                   as Trustee

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

                         SECOND SUPPLEMENTAL INDENTURE

                          Dated as of December 1, 2003

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

Supplement to Indenture dated as of January 30, 1998, by and between TCA Cable
TV, Inc. and JPMorgan Chase Bank (as successor to Chase Bank of Texas, National
Association), as Trustee, as amended by the First Supplemental Indenture dated
as of August 12, 1999, by and among TCA Cable TV, Inc., Cox Classic Cable, Inc.
and JPMorgan Chase Bank (as successor to Chase Bank of Texas, National
Association), as Trustee, relating to the Company's Securities

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<PAGE>


                         SECOND SUPPLEMENTAL INDENTURE


         SECOND SUPPLEMENTAL INDENTURE, dated as of December 1, 2003, among Cox
Classic Cable, Inc., a corporation duly organized and existing under the laws
of the State of Delaware (the "Company"), CoxCom, Inc., a corporation duly
organized and existing under the laws of the State of Delaware ("Cox"), and
JPMorgan Chase Bank (as successor to Chase Bank of Texas, National Association)
(the "Trustee"), as Trustee under the Indenture hereinafter mentioned.

                                   WITNESSETH

         WHEREAS, TCA Cable TV, Inc. ("TCA") heretofore executed and delivered
to the Trustee an Indenture dated as of January 30, 1998 (the "Initial
Indenture");

         WHEREAS, on or about February 5, 1998, TCA issued and sold
$200,000,000 aggregate principal amount of 6.53% Debentures due 2028 under the
Initial Indenture;

         WHEREAS, TCA and the Company heretofore executed and delivered to the
Trustee a First Supplemental Indenture dated as of August 12, 1999 (the "First
Supplemental Indenture," and together with the Initial Indenture, the
"Indenture"), providing that the Company would agree to be bound by all the
terms, conditions and covenants of the Initial Indenture and assume all of
TCA's obligations under the Initial Indenture;

         WHEREAS, pursuant to that certain Agreement and Plan of Merger dated
as of December 1, 2003, by and among the Company and Cox, the Company will be
merged with and into Cox and will cease to exist (the "Restructuring");

         WHEREAS, Section 901 of the Indenture, "Supplemental Indentures
Without Consent of Securityholders," provides that provisions of the Indenture
may be amended or supplemented without the consent of the Holders with respect
to certain matters therein identified;

         WHEREAS, Cox desires in and by this Second Supplemental Indenture to
be bound by all the terms and conditions of the Indenture and assume all of the
Company's obligations under the Indenture; and

         WHEREAS, all conditions necessary to authorize the execution and
delivery of this Second Supplemental Indenture and to make this Second
Supplemental Indenture valid and binding have been complied with or have been
done or performed;

         NOW THEREFORE, in consideration of the above premises, and in order to
comply with the terms of the Indenture, the Company and Cox covenant with the
Trustee as follows:


<PAGE>
                                  ARTICLE ONE
                                  DEFINITIONS

         Section 1.01. For all purposes of the Indenture and this Second
Supplemental Indenture, except as otherwise expressly provided or unless the
context otherwise requires:

         (a)      the words "herein", "hereof" and "hereunder" and other words
of similar import refer to the Indenture and this Second Supplemental Indenture
as a whole and not to any particular Article, Section or subdivision; and

         (b)      capitalized terms used but not defined herein shall have the
meanings assigned to them in the Indenture.

                                  ARTICLE TWO
                            AMENDMENT AND SUPPLEMENT

         Section 2.01. Cox hereby agrees to be bound by all of the terms,
conditions and covenants of the Indenture and assumes all of the Company's
obligations under the Indenture and the Securities issued thereunder on the
terms and conditions stated therein.

         Section 2.02. The Company and Cox jointly represent and warrant that
immediately after giving effect to the Restructuring, no Event of Default, and
no event which, after notice or lapse of time, or both, would become an Event
of Default, shall have happened and be continuing.

                                                    ARTICLE THREE
                                                    MISCELLANEOUS

         Section 3.01. All of the terms and conditions of the Indenture shall
remain in full force and effect, except as modified hereby, and the Holders are
bound by the Indenture, as amended, pursuant to Section 904 thereof.

         Section 3.02. The Trustee accepts the modification of the Indenture
effected by this Second Supplemental Indenture, but only upon the terms and
conditions set forth in the Indenture. Without limiting the generality of the
foregoing, the Trustee assumes no responsibility for the correctness of the
recitals herein contained, which shall be taken as the statements of the
Company. The Trustee makes no representation and shall have no responsibility
as to the validity of this Second Supplemental Indenture.

         Section 3.03. In case any provision in this Second Supplemental
Indenture shall be invalid, illegal or unenforceable, the validity, legality or
enforceability of the remaining provisions of this Second Supplemental
Indenture or the Indenture shall not in any way be affected or impaired
thereby.


                                       2
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         Section 3.04. This Second Supplemental Indenture shall be deemed to be
a contract made under the laws of the State of New York and for all purposes
shall be governed by and construed in accordance with the laws of the State of
New York without regard to principles of conflicts of laws.

         Section 3.05. This Second Supplemental Indenture may be executed in
any number of counterparts, each of which when so executed shall be deemed to
be an original, but such counterparts shall together constitute but one and the
same instrument.

         Section 3.06. This Second Supplemental Indenture shall be effective
simultaneously with the effectiveness of the Restructuring, which shall occur
on December 1, 2003.


                           [Signature Page Follows.]


                                       3
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         IN WITNESS WHEREOF, the Company, Cox and the Trustee have caused their
names to be signed hereto by their respective officers thereunder duly
authorized and their respective corporate seals, duly attested, to be hereunto
duly affixed, all as of the day and the year first above written.


                                        COX CLASSIC CABLE, INC.


[CORPORATE SEAL]


                                        By: /s/ Susan W. Coker
                                            -----------------------------------
                                            Name: Susan W. Coker
Attest: /s/ Andrew A. Merdek                Title: Treasurer
        ---------------------------
        Andrew A. Merdek, Secretary


                                        COXCOM, INC.


[CORPORATE SEAL]


                                        By: /s/ Jimmy W. Hayes
                                            -----------------------------------
                                            Name: Jimmy W. Hayes
Attest: /s/ Andrew A. Merdek                Title: Vice President
        ---------------------------
        Andrew A. Merdek, Secretary


                                        JPMorgan Chase Bank
                                        (as successor to CHASE BANK OF TEXAS,
                                        National Association), as Trustee


[CORPORATE SEAL]


                                        By: /s/ Carol Logan
                                            -----------------------------------
                                            Name: Carol Logan
Attest: [illegible]                         Title: Vice President and
        ---------------------------                Trust Officer