(abridged) We present a catalog of 2357 point sources detected during 590 ks of Chandra observations of the 17-by-17 arcminute field around Sgr A*. This field encompasses a physical area of 40 by 40 pc at a distance of 8 kpc. The completeness limit of the sample at the Galactic center is 10^{31} erg s^{-1} (2.0--8.0 keV), while the detection limit is an order of magnitude lower. The 281 sources detected below 1.5 keV are mainly in the foreground of the Galactic center, while comparisons to the Chandra deep fields at high Galactic latitudes suggest that only about 100 of the observed sources are background AGN. The surface density of absorbed sources (not detected below 1.5 keV) falls off as 1/theta away from Sgr A*, in agreement with the distribution of stars in infrared surveys. Point sources brighter than our completeness limit produce 10% of the flux previously attributed to diffuse emission. The log(N)-log(S) distribution of the Galactic center sources is extremely steep (power-law slope alpha = 1.7). If this distribution extends down to a flux of 10^{-17} erg cm^{-1} s^{-1} (10^{29} erg s^{-1} at 8 kpc, 2.0--8.0 keV) with the same slope, then point sources would account for all of the previously reported diffuse emission. Therefore, the 2.0--8.0 keV luminosity distribution must flatten between 10^{29} - 10^{31} erg s^{-1}. Finally, the spectra of more than half of the Galactic center sources are very hard, and can be described by a power law ($E^{-Gamma}) with photon index Gamma < 1. Such hard spectra have been seen previously only from magnetically accreting white dwarfs and wind-accreting neutron stars, suggesting that there are large numbers of these systems in our field.
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