We report measurements with the Very Long Baseline Array of the proper motion of Sgr A* relative to two extragalactic radio sources spanning 18 years. The apparent motion of Sgr A* is -6.411 +/- 0.008 mas/yr along the Galactic plane and -0.219 +/- 0.007 mas/yr 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 in the direction of Galactic rotation and -0.85 +/- 0.75 km/s 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 x 10^6 Msun deduced from stellar orbits. The intrinsic size of Sgr A* is comparable to its Schwarzschild radius, and the implied mass density of >4 x 10^23 Msun/pc^-3 very close to that expected for a black hole, providing overwhelming evidence that it is indeed a super-massive black hole. Finally, the existence of "intermediate-mass" black holes more massive than 3 x 10^4 Msun between approximately 0.003 and 0.1 pc from Sgr A*are excluded. (read more)