We report measurements with the Very Long Baseline Array of the proper motion of Sgr A* relative to two extragalactic radio sources spanning 18 yr. The apparent motion of Sgr A* is -6.411 ± 0.008 mas yr-1 along the Galactic plane and -0.219 ± 0.007 mas yr-1 toward the North Galactic Pole. This apparent motion can almost entirely be attributed to the effects of the Sun's orbit about the Galactic center. Removing these effects yields residuals of -0.58 ± 2.23 km s-1 in the direction of Galactic rotation and -0.85 ± 0.75 km s-1 toward the North Galactic Pole. A maximum-likelihood analysis of the motion, both in the Galactic plane and perpendicular to it, expected for a massive object within the Galactic center stellar cluster indicates that the radiative source, Sgr A*, contains more than about 25% of the gravitational mass of 4 × 106 M☉ deduced from stellar orbits. The intrinsic size of Sgr A* is comparable to its Schwarzschild radius, and the implied mass density of ≳ 4× 1023 M☉ pc-3 is very close to that expected for a black hole, providing overwhelming evidence that it is indeed a supermassive black hole. Finally, "intermediate mass" black holes more massive than ≍3 × 104 M☉ between approximately 0.003 and 0.1 pc from Sgr A* are excluded.