
<PAGE>
 
                                                                    EXHIBIT 99.5

                      SECOND AMENDMENT TO CREDIT AGREEMENT
                      ------------------------------------

          THIS SECOND AMENDMENT TO CREDIT AGREEMENT (this "Second Amendment") is
made and entered into as of this 29th day of March, 1996 by and among ACC CORP.,
a corporation organized under the laws of Delaware ("ACC"), the Subsidiaries of
ACC (the "Subsidiaries" and, together with ACC, the "Borrowers") party to the
Credit Agreement (as defined below), the Lenders party to the Credit Agreement,
FIRST UNION NATIONAL BANK OF NORTH CAROLINA, a national banking association
("First Union" or the "Administrative Agent"), as Managing Agent and
Administrative Agent and FLEET NATIONAL BANK OF CONNECTICUT, N.A. (formerly
known as Shawmut Bank Connecticut, N.A.), a national banking association ("Fleet
Bank" and, together with First Union, the "Agents"), as Managing Agent.



                              Statement of Purpose
                              --------------------

          The Lenders have extended certain credit facilities to the Borrowers
pursuant to the Credit Agreement dated July 21, 1995 among the Borrowers, the
Lenders and the Agents, as amended by the First Amendment to Credit Agreement
dated as of October 31, 1995 (as amended, the "Credit Agreement").  The parties
now desire to further amend certain provisions of the Credit Agreement on the
terms and conditions set forth below.

          NOW THEREFORE, for good and valuable consideration, the receipt and
adequacy of which is hereby acknowledged, the parties hereto agree as follows:

          1.  Effect of Amendment.  Except as expressly amended hereby, the
              -------------------                                          
Credit Agreement and the other Loan Documents shall be and remain in full force
and effect.

          2.  Capitalized Terms.  All capitalized undefined terms used in this
              -----------------                                               
Second Amendment shall have the meanings assigned thereto in the Credit
Agreement.

          3.  Modification of Credit Agreement.  The Credit Agreement shall 
              --------------------------------      
hereby be amended as follows:

          (a) Hedging Agreements.  Section 7.13 of the Credit Agreement shall
              ------------------                                             
hereby be deleted in its entirety and the following Section 7.13 shall be
substituted in lieu thereof:

               "SECTION 7.13  Hedging Agreements.  (a) From and after June 30,
                              ------------------
     1996, maintain at all times Hedging Agreements with respect to interest 
     rate exposure under the Credit Agreement with durations of at least two 
     years from the respective
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     dates of execution thereof and an aggregate notional principal amount
     thereunder equal to at least fifty percent (50%) of the aggregate principal
     amount of outstanding Loans at interest rates not to exceed two percent
     (2%) over the then current three month LIBOR Rate at the time of execution
     of such Hedging Agreements with respect to the applicable Permitted
     Currency and otherwise in form and substance reasonably satisfactory to the
     Managing Agents and (b) maintain at all times Hedging Agreements with
     respect to currency risk in form and substance reasonably satisfactory to
     the Managing Agents."

     (b) Fixed Charge Coverage Ratio.  Section 8.3 of the Credit Agreement shall
         ---------------------------                                            
hereby be deleted in its entirety and the following Section 8.3 shall be
substituted in lieu thereof:

          "SECTION 8.3  Fixed Charge Coverage Ratio.  As of any date of
                        ---------------------------                    
     determination, permit the ratio of (a) (i), for any calculation period
     through June 30, 1996, Operating Cash Flow for the two (2) consecutive
     fiscal quarters ending on or immediately prior to such date times two (2),
                                                                 -----         
     and for any calculation period thereafter, Operating Cash Flow for the
     period of four (4) consecutive fiscal quarters ending on or immediately
     prior to such date to (b) Fixed Charges for such period, to be less than
     the corresponding ratio set forth below:
<TABLE>
<CAPTION>
 
                 Period               Ratio
                 ------               -----     
          <S>                        <C>
 
          Closing date through
            September 30, 1995       .30 to 1.0
          October 1, 1995 through
            September 30, 1996       .50 to 1.0
          October 1, 1996 through
            December 31, 1996        .60 to 1.0
          January 1, 1997 and
            thereafter               1.15 to 1.0
</TABLE>

      4.  Conditions Precedent.  The effectiveness of the amendments set forth
          --------------------                                                
herein shall be conditioned upon delivery by the Borrowers to the Agent of an
amendment fee in an amount equal to $87,500 less any and all expenses due
                                            ----                         
pursuant to Paragraph 6 below, to be distributed by the Agent among the Lenders
in accordance with their respective Commitment Percentages.

      5.  Representations and Warranties/No Default.  By their execution hereof,
          -----------------------------------------                             
the Borrowers hereby certify that each of the representations and warranties set
forth in the Credit Agreement and the other Loan Documents is true and correct
as of the date hereof as if fully set forth herein and that as of the date
hereof no Default or Event of Default has occurred and is continuing.

                                       2
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      6.  Expenses.  The Borrowers shall pay all reasonable out-of-pocket
          --------                                                       
expenses of the Agents in connection with the preparation, execution and
delivery of this First Amendment, including without limitation, the reasonable
fees and disbursements of counsel for the Agents.

     7.   Governing Law.  This Second Amendment shall be governed by and
          -------------                                                 
construed in accordance with the laws of the State of North Carolina.

     8.   Counterparts.  This Second Amendment may be executed in separate
          ------------                                                    
counterparts, each of which when executed and delivered is an original but all
of which taken together constitute one and the same instrument.

                                       3
<PAGE>
 
     IN WITNESS WHEREOF, the parties hereto have caused this Second Amendment to
be duly executed as of the date and year first above written.

 
[CORPORATE SEAL]              ACC CORP.


                              By:   /s/ John J. Zimmer
                                  -------------------------------
                                 Name:  John J. Zimmer
                                       --------------------------
                                 Title:  Vice President - Finance
                                       --------------------------


[CORPORATE SEAL]              ACC LONG DISTANCE CORP.


                              By:   /s/ John J. Zimmer
                                  -------------------------------
                                 Name:  John J. Zimmer
                                       --------------------------
                                 Title:  Controller
                                        -------------------------


[CORPORATE SEAL]              ACC NATIONAL TELECOM CORP.

                              By:   /s/ John J. Zimmer
                                  -------------------------------
                                 Name:  John J. Zimmer
                                       --------------------------
                                 Title:  Controller
                                        -------------------------
 


[CORPORATE SEAL]              ACC LONG DISTANCE OF MASSACHUSETTS CORP.


                              By:   /s/ John J. Zimmer
                                  -------------------------------
                                 Name:  John J. Zimmer
                                       --------------------------
                                 Title:  Controller
                                        -------------------------


[CORPORATE SEAL]              ACC RADIO CORP.

                              By:   /s/ John J. Zimmer
                                  -------------------------------
                                 Name:  John J. Zimmer
                                       --------------------------
                                 Title:  Controller
                                        -------------------------

                                       4
<PAGE>
 
[CORPORATE SEAL]              ACC NATIONAL LONG DISTANCE CORP.


                              By:   /s/ John J. Zimmer
                                  -------------------------------
                                 Name:  John J. Zimmer
                                       --------------------------
                                 Title:  Controller
                                        -------------------------

[CORPORATE SEAL]              ACC LONG DISTANCE U.K., LTD.


                              By:   /s/ John J. Zimmer
                                  -------------------------------
                                 Name:  John J. Zimmer
                                       --------------------------
                                 Title:  Attorney-in-fact
                                        -------------------------


                              FIRST UNION NATIONAL BANK OF NORTH 
                              CAROLINA, as Administrative Agent, 
                              Managing Agent and Lender

                              By:   /s/ James Redmond
                                  -------------------------------
                                 Name:  James Redmond
                                       --------------------------
                                 Title:  Senior Vice President
                                        -------------------------


                              FLEET NATIONAL BANK OF CONNECTICUT, 
                              N.A. (formerly known as Shawmut 
                              Bank Connecticut, N.A.), as
                              Managing Agent and Lender

                              By:   /s/ Wendy E. Klepper
                                  -------------------------------
                                 Name:  Wendy E. Klepper
                                       --------------------------
                                 Title:  Assistant Vice President
                                        -------------------------

                                       5
