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We show that observations by Backer and collaborators over the past twodecades constrain the time derivative of the proper motion of Sgr A* to be lessthan 0.14 mas yr^{-2}. Using this result and a preliminary measurement byEckart and Genzel of sigma ~500 kms for the velocity dispersion of the starcluster within 0.2" of Sgr A*, we derive the following implications. First, ifthe nuclear star cluster is dominated by a massive black hole, then either SgrA* is that black hole or it orbits the black hole with a radius less than 3 AU.Second, even if the star cluster does not contain a massive black hole, Sgr A*is constrained to move slower than 20 kms (1 sigma) relative to the center ofmass of the cluster. The Galactocentric distance is therefore R_0=7.5 +/- 0.7kpc, independent of the nature of Sgr A*. These error bars could besubstantially reduced by future observations. If they are, it will also bepossible to probe the motion of the nuclear star cluster relative to the centerof mass of the Galaxy at the ~4 kms level.Comment: 9 pages, no figure

Nonacceleration of Sagittarius A*: Implications for Galactic Structure