We examine the spectrum of diffuse emission detected in the 17' by 17' field around \sgrastar during 625 ks of \chandra observations. The spectrum exhibits He-like and H-like lines from Si, S, Ar, Ca, and Fe, that are consistent with originating in a two-temperature plasma, as well as a prominent low-ionization Fe K- alpha line. The cooler, kT 0.8 keV plasma differs in surface brightness across the image between (0.2 - 1.8)*10-13 \ergcmsarcmin (observed, 2-8 keV). This soft plasma is probably heated by supernovae, along with a small contribution from the winds of massive Wolf-Rayet and O stars. The radiative cooling rate of the soft plasma within the inner 20 pc of the Galaxy could be balanced by 1% of the kinetic energy of one supernova every 3*105 y. The hotter, kT 8 keV component is more spatially uniform, with a surface brightness of (1.5-2.6)*10-13 \ergcmsarcmin (observed; 2-8) keV. The intensity of the hard plasma is correlated with that of the soft, but they are probably only indirectly related, because neither supernova remnants nor WR/O stars are observed to produce thermal plasma hotter than kT 3 keV. Moreover, a kT 8 keV plasma would be too hot to be bound to the Galactic center, and therefore would form a slow wind or fountain of plasma. The energy required to sustain such a freely-expanding plasma within the inner 20 pc of the Galaxy is 1040 \ergsec. This corresponds to the entire kinetic energy of one supernova every 3000 y, which is unreasonably high. However, alternative explanations for the kT 8 keV diffuse emission are equally unsatisfying. The hard X-rays are unlikely to result from undetected point sources, because no known population of stellar object is numerous enough to the observed surface brightness. There is also no evidence that non-thermal mechanisms for producing the hard emission are operating, as the expected shifts in the line energies and ratios from their collisional equilibrium values are not observed. We are left to conclude that either there is a significant shortcoming in our understanding of the mechanisms that heat the interstellar medium, or that a population of faint (< 1031 erg s-1), hard X-ray sources that are a factor of 10 more numerous than CVs remains to be discovered.