ABSTRACT: We observed SgrA* and two extragalactic radio sources with the VLBA over a period of two years. Relative positions of the sources were measured with an accuracy approaching 0.0001 arcsec. The apparent proper motion of SgrA* was determined to be 0.0060 +/- 0.0004 arcsec per year, almost entirely in the plane of the Galaxy. The effects of the orbit of the Sun around the galactic center can account for our observations, and any residual proper motion of SgrA*, with respect to extragalactic sources, is less than about 20 km/sec. Our results and those of Backer and Sramek, from observations with the VLA, appear consistent, making the observational constraints robust. The large observed speeds of stars in the inner region of the dense central cluster surrounding SgrA*, coupled with the very slow speed of SgrA*, yields a lower limit to the mass of SgrA*. If SgrA* is a massive object, dominating the central gravitational potential, then equipartition of kinetic energy cannot be used to place a limit on the mass of SgrA* from stellar motions in the central cluster. Instead, direct gravitational interactions dominate, and simple arguments based on conservation of angular momentum (as well as N-body simulations) yield a lower limit on the mass of SgrA* of order 1,000 solar masses. While the low luminosity of SgrA* might possibly have come from, for example, a contact binary containing of order 10 solar masses and radiating near its Eddington limit, the proper motions rule out a "stellar" origin for SgrA*. Current observations are consistent with SgrA* containing all of the nearly 2.6 million solar masses, inferred from the stellar motions, in the form of massive black hole and radiating at less than 0.1 percent of the Eddington limit.