  We report measurements with the VLBA of the position of Sgr A* with respect
to two extragalactic radio sources over a period of eight years. The apparent
proper motion of \sgrab relative to J1745-283 is 6.379 +/- 0.024 mas/y along a
position angle of 209.60 +/- 0.18 degrees, 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. 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
approximately 4 x 10^6 Msun 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 ~10^22 Msun/pc^3 is within about three orders of
magnitude of a comparable super-massive 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 10^6 Msun and
provides overwhelming evidence that it is in the form of a super-massive black
hole. Finally, the existence of "intermediate mass" black holes more massive
than ~10^4 Msun between roughly 10^3 and 10^5 AU from Sgr A* are excluded.
