We report the results of a 48,000 s observation of the central approximately 34' of M31 with the ROSAT HRI. We detect 86 X-ray sources with luminosities above approximately 1.4 x 10(36) ergs s-1 (3.5 x 10(-4) counts s-1). Eighteen sources are identified with globular clusters and two with supernova remnants. A point source coincident with the optical and radio nucleus is also detected. The cumulative luminosity distribution for all sources within 5' (approximately 1 kpc) of the nucleus flattens below approximately 2 x 10(37) ergs s-1. An extrapolation of this distribution to fainter luminosities indicates that only approximately 15%-26% of the diffuse X-ray emission in the bulge can be attributed to faint unresolved sources from this distribution. Estimates for contributions from other, less luminous X-ray source populations cannot account for the remainder, and we conclude that much of the unresolved X-ray emission in the bulge is either truly diffuse or due to a new class of X-ray sources. A comparison of the current observations with Einstein observations made 10 years ago indicates that approximately 42% of the sources within 7.5' of the nucleus are variable. The spatial distribution and number density of X-ray sources in M31 appear distinctly different from those in our Galaxy.