We show that observations by Backer and collaborators over the past two decades constrain the time derivative of the proper motion of Sgr A* to be less than 0.14 mas yr^-2. Using this result and a preliminary measurement by Eckart and Genzel of sigma 500 kms for the velocity dispersion of the star cluster within 0.2'' of Sgr A*, we derive the following implications. First, if the nuclear star cluster is dominated by a massive black hole, then either Sgr A* 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 of mass of the cluster. The Galactocentric distance is therefore R_0=7.5 +/- 0.7 kpc, independent of the nature of Sgr A*. These error bars could be substantially reduced by future observations. If they are, it will also be possible to probe the motion of the nuclear star cluster relative to the center of mass of the Galaxy at the 4 kms level.