We present results of our Chandra observation with ACIS-I centered on the position of Sagittarius A* (Sgr A*), the compact nonthermal radio source associated with the massive black hole (MBH) at the dynamical center of the Milky Way Galaxy. We have obtained the first high spatial resolution ( 1''), hard X-ray (0.5-7 keV) image of the central 40 pc (17') of the Galaxy. We have discovered an X-ray source, \objectname[]CXOGC J174540.0-290027, coincident with the radio position of Sgr A* to within 0.''35, corresponding to a maximum projected distance of 16 light-days for an assumed distance to the center of the Galaxy of 8.0 kpc. We received 222+/-17 (1 sigma ) net counts from the source in 40.3 ks. The source is detected with high significance, S/N 37 sigma , despite the highly elevated diffuse X-ray background in the central parsec of the Galaxy. Due to the low number of counts, the spectrum is well fit either by an absorbed power-law model with photon index \Gamma = 2.7+1.3_-0.9 (N(E) \propto E-\Gamma photons cm-2 s-1 keV-1) and column density N_H = (9.8+4.4_-3.0) * 1022 cm-2 (90% confidence interval) or by an absorbed optically thin thermal plasma model with kT = 1.9+0.9_-0.5 keV and N_H = (11.5+4.4_-3.1) * 1022 cm-2. Using the power-law model, the measured (absorbed) flux in the 2-10 keV band is (1.3+0.4_-0.2) * 10-13 ergs cm-2 s-1, and the absorption-corrected luminosity is (2.4+3.0_-0.6) * 1033 ergs s-1. The X-ray source coincident with Sgr A* is resolved, with an apparent diameter of 1''. We report the possible detection, at the 2.7 sigma significance level, of rapid continuum variability on a timescale of several hours. We also report the possible detection of an Fe K alpha line at the 2 sigma level. The long-term variability of Sgr A* is constrained via comparison with the Rosat/PSPC observation in 1992. The origin of the X-ray emission (MBH vs. stellar) and the implications of our observation for the various proposed MBH emission mechanisms are discussed. The current observations, while of limited signal-to-noise, are consistent with the presence of both thermal and nonthermal emission components in the Sgr A* spectrum. We also briefly discuss the complex structure of the X-ray emission from the Sgr A radio complex and along the Galactic plane and present morphological evidence that Sgr A* and Sgr A West lie within the hot plasma in the central cavity of Sgr A East. Over 150 point sources are detected in the 17' * 17' field of view. Our survey of X-ray sources is complete down to a limiting 2-10 keV absorbed flux of F_X 1.7 * 10-14 ergs cm-2 s-1. For sources at the distance of the Galactic Center, the corresponding absorption-corrected luminosity is L_ X 2.5 * 1032 ergs s-1. The complete flux-limited sample contains 85 sources. Finally, we present an analysis of the integrated emission from the detected point sources and the diffuse emission within the central 0.4 pc (10'') of the Galaxy.