Over a two year period, we have conducted a diffraction-limited imaging study at 2.2 mu m of the inner 6\tt'' * 6\tt'' of the Galaxy's central stellar cluster using the W. M. Keck 10-m telescope. The K band images obtained in 1995 June, 1996 June, and 1997 May have the highest angular resolution obtained at near-infrared wavelengths from ground or space ( theta _res = 0.'' 05 = 0.002 pc) and reveal a large population of faint stars. We use an unbiased approach for identifying and selecting stars to be included in this proper motion study, which results in a sample of 90 stars with brightness ranging from K = 9 to 17 mag and two-dimensional velocities as large as 1,400 +/- 100 km/sec. Compared to earlier work (Eckart et al. 1997; Genzel et al. 1997), the source confusion is reduced by a factor of 9, the number of stars with proper motion measurement in the central 25 arcsec2 of our galaxy is doubled, and the accuracy of the velocity measurements in the central 1 arcsec2 is improved by a factor of 4. The peaks of both the stellar surface density and the velocity dispersion are consistent with the position of the unusual radio source and black hole candidate, Sgr A*, suggesting that Sgr A* is coincident (+/- 0.'' 1) with the dynamical center of the Galaxy. As a function of distance from Sgr A*, the velocity dispersion displays a falloff well fit by Keplerian motion ( sigma v ~ r-0.5 +/- 0.1) about a central dark mass of 2.6 (+/- 0.2) * 106 M_o confined to a volume of at most 10-6 pc3, consistent with earlier results. Although uncertainties in the measurements mathematically allow for the matter to be distributed over this volume as a cluster, no realistic cluster is physically tenable. Thus, independent of the presence of Sgr A*, the large inferred central density of at least 1012 M_o / pc3, which exceeds the volume-averaged mass densities found at the center of any other galaxy, leads us to the conclusion that our Galaxy harbors a massive central black hole.