We report on the X-ray emission from the shell-like, non-thermal radio source Sgr A East (SNR 000.0+00.0) located in the inner few parsecs of the Galaxy based on observations made with the ACIS detector on board the Chandra X-ray Observatory. This is the first time Sgr A East has been clearly resolved from other complex structures in the region. The X-ray emitting region is concentrated within the central 2 pc of the larger radio shell. The spectrum shows strong K alpha lines from highly ionized ions of S, Ar, Ca, and Fe. A simple isothermal plasma model gives electron temperature 2 keV, absorption column 1 * 1023 H cm-2, luminosity 8 * 1034 ergs s-1 in the 2-10 keV band, and gas mass 2\eta\frac12 M_o with a filling factor \eta. The plasma appears to be rich in heavy elements, over-abundant by roughly a factor of four with respect to solar abundances, and shows a spatial gradient of elemental abundance: the spatial distribution of iron is more compact than that of the lighter elements. The gas mass and elemental abundance of the X-ray emission support the long-standing hypothesis that Sgr A East is a supernova remnant (SNR), maybe produced by the Type II supernova explosion of a massive star with a main-sequence mass of 13-20 Mo. The combination of the radio and X-ray morphologies classifies Sgr A East as a new metal-rich ``mixed morphology'' (MM) SNR. The size of the Sgr A East radio shell is the smallest of the known MM SNRs, which strongly suggests that the ejecta have expanded into a very dense interstellar medium. The ejecta-dominated chemical compositions of the plasma indicate that the ambient materials should be highly homogeneous. We thus evaluate a simplified dynamical evolution model where a SNR was formed about 10,000 years ago and expanded into an ambient medium with a homogeneous density of 103 cm-3. The model roughly reproduces most of the observed properties in the X-ray and radio wavelengths. A comparison with the radio observations requires the dense ambient medium to be ionized, but a luminous X-ray irradiator with an expected X-ray luminosity of 1040 ergs s-1 is not currently present. The presence of the ionized gas may be explained if the massive black hole (MBH) associated with the compact, non-thermal radio source Sgr A^* was bright in X-rays about three hundred years ago, but is presently dim. It is possible that the dust/molecular ridge compressed by the forward shock of Sgr A East hit Sgr A* in the past, and the passage of the ridge may have supplied material to accrete onto the black hole in the past, and may have removed material from the black hole vicinity, leading to its present quiescent state. This may be a specific example of the intimate relationship between a SNR and massive black hole accretion activity in galactic nuclei.