Proper motions and radial velocities of luminous infrared stars in the galactic center have provided strong evidence for a dark mass of 2.5* 106 \Msun in the central 0.05 pc of the galaxy. The leading hypothesis for this mass is a black hole. High angular resolution measurements at radio wavelengths find a compact radio source, Sagittarius (Sgr) A^*, that is either the faint glow from a small amount of material accreting onto the hole with low radiative efficiency or a miniature AGN core-jet system. We provide in this paper a full report on the first program that has measured the apparent proper motion of Sgr A^* with respect to background extragalactic reference frame. Our current result is: mu _l,*=[-6.18+/- 0.19] mas y-1 mu _b,*=[-0.65+/- 0.17] mas y-1. The observations were obtained with the NRAO Very Large Array at 4.9 GHz over sixteen years. The proper motion of Sgr A^* provides an estimate of its mass based on equipartition of kinetic energy between the hole and the surrounding stars. The measured motion is largest in galactic longitude. This component of the motion is consistent with the secular parallax that results from the rotation of the solar system about the center, which is a global measure of Oort's constants (A-B), with no additional peculiar motion of Sgr A^*. The current uncertainty in Oort's galactic rotation constants limits the use of this component of the proper motion for a mass inference. In latitude we find a small, and weakly significant, peculiar motion of Sgr A^*, -19+/- 7 km s-1 after correction for the motion of the solar system with respect to the local standard of rest. We consider sources of peculiar motion of Sgr A^* ranging from unstable radio wave propagation through intervening turbulent plasma to the effects of asymmetric masses in the center. These fail to account for a significant peculiar motion. One can appeal to an m=1 dynamical instability that numerical simulations have revealed. However, the measurement of a latitude peculiar proper motion of comparable magnitude and error but with opposite sign in the companion paper by \citeauthorReid99 (\citeyearReid99) leads us to conclude at the present time that our errors may be underestimated, and that the actual peculiar motion might therefore be closer to zero. Improvement of these measurements with further observations and resolving the differences between independent experiments will provide the accuracies of a few km s-1 in both coordinates that will provide both a black hole mass estimate and a definitive determination of Oort's galactic rotation constants on a global galactic scale.