We observed Sgr A* and two extragalactic radio sources nearby in angle with the VLBA over a period of two years and measured relative positions with an accuracy approaching 0.1 mas. The apparent proper motion of Sgr A* relative to J1745-283 is 5.90+/-0.4 mas /yr, almost entirely in the plane of the Galaxy. The effects of the orbit of the Sun around the Galactic Center can account for this motion, and any residual proper motion of Sgr A*, with respect to extragalactic sources, is less than about 20 km/sec . Assuming that Sgr A* is at rest at the center of the Galaxy, we estimate that the circular rotation speed in the Galaxy at the position of the Sun, THETA 0, is 219+/-20 km/s, scaled by R0/8.0 kpc. Current observations are consistent with Sgr A* containing all of the nearly 2.6*106 Mo, deduced from stellar proper motions, in the form of a massive black hole. While the low luminosity of Sgr A*, for example, might possibly have come from a contact binary containing of order 10 Mo, the lack of substantial motion rules out a ``stellar'' origin for Sgr A*. The very slow speed of Sgr A* yields a lower limit to the mass of Sgr A* of about 1,000 Mo. Even for this mass, Sgr A* appears to be radiating at less than 0.1% of its Eddington limit.