We report measurements with the Very Long Baseline Array (VLBA) of the position of Sgr A* with respect to two extragalactic radio sources over a period of 8 yr. The apparent proper motion of Sgr A* relative to J1745-283 is 6.379 ± 0.024 mas yr^-1 along a position angle of 20960 ± 018, 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 the residual proper motion of Sgr A* perpendicular to the plane of the Galaxy is -0.4 ± 0.9 km s^-1. A maximum likelihood analysis of the motion expected for a massive object within the observed Galactic center stellar cluster indicates that Sgr A* contains more than about 10% of the ≈4 × 10⁶ M deduced from stellar orbits. The intrinsic size of Sgr A*, as measured by several investigators, is less than 1 AU, and the implied mass density of ~10²² M pc^-3 is within about 3 orders of magnitude of a comparable supermassive black hole within its Schwarzschild radius. Our observations provide the first direct evidence that a compact radiative source at the center of a galaxy contains of order 10⁶ M and provides overwhelming evidence that it is in the form of a supermassive black hole. Finally, the existence of "intermediate-mass" black holes more massive than ~10⁴ M between roughly 10³ and 10⁵ AU from Sgr A* is excluded.