SECTION 906 CERTIFICATION

                Certification Pursuant to 18 U.S.C. Section 1350,
                             As Adopted Pursuant to
                  Section 906 of the Sarbanes-Oxley Act of 2002

Morgan Stanley New York Municipal Money Market Trust

     In connection with the Report on Form N-CSR (the "Report") of the
above-named issuer for the period ended June 30, 2006 that is accompanied by
this certification, the undersigned hereby certifies that:

1.   The Report fully complies with the requirements of Section 13(a) or 15(d)
     of the Securities Exchange Act of 1934; and

2.   The information contained in the Report fairly presents, in all material
     respects, the financial condition and results of operations of the Issuer.

Date: August 10, 2006


                                        /s/ Ronald E. Robison
                                        ----------------------------------------
                                        Ronald E. Robison
                                        Principal Executive Officer

A signed original of this written statement required by Section 906 has been
provided to Morgan Stanley New York Municipal Money Market Trust and will be
retained by Morgan Stanley New York Municipal Money Market Trust and furnished
to the Securities and Exchange Commission or its staff upon request.


                                        8



                            SECTION 906 CERTIFICATION

                Certification Pursuant to 18 U.S.C. Section 1350,
                             As Adopted Pursuant to
                  Section 906 of the Sarbanes-Oxley Act of 2002

Morgan Stanley New York Municipal Money Market Trust

     In connection with the Report on Form N-CSR (the "Report") of the
above-named issuer for the period ended June 30, 2006 that is accompanied by
this certification, the undersigned hereby certifies that:

1.   The Report fully complies with the requirements of Section 13(a) or 15(d)
     of the Securities Exchange Act of 1934; and

2.   The information contained in the Report fairly presents, in all material
     respects, the financial condition and results of operations of the Issuer.

Date: August 10, 2006


                                        /s/ Francis Smith
                                        ----------------------------------------
                                        Francis Smith
                                        Principal Financial Officer

A signed original of this written statement required by Section 906 has been
provided to Morgan Stanley New York Municipal Money Market Trust and will be
retained by Morgan Stanley New York Municipal Money Market Trust and furnished
to the Securities and Exchange Commission or its staff upon request.


                                        9