  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.
