We present the results of a comprehensive Keck/DEIMOS spectroscopic survey of the ultra-faint Milky Way satellite galaxy Segue 1. We have obtained velocity measurements for 98.2% of the stars within 67 pc (10 arcmin, or 2.3 half-light radii) of the center of Segue 1 that have colors and magnitudes consistent with membership, down to a magnitude limit of r=21.7. Based on photometric, kinematic, and metallicity information, we identify 71 stars as probable Segue 1 members, including some as far out as 87 pc. After correcting for the influence of binary stars using repeated velocity measurements, we determine a velocity dispersion of 3.7^{+1.4}_{-1.1} km/s, with a corresponding mass within the half-light radius of 5.8^{+8.2}_{-3.1} x 10^5 Msun. The stellar kinematics of Segue 1 require very high mass-to-light ratios unless the system is far from dynamical equilibrium, even if the period distribution of unresolved binary stars is skewed toward implausibly short periods. With a total luminosity less than that of a single bright red giant and a V-band mass-to-light ratio of 3400 Msun/Lsun, Segue 1 is the darkest galaxy currently known. We critically re-examine recent claims that Segue 1 is a tidally disrupting star cluster and that kinematic samples are contaminated by the Sagittarius stream. The extremely low metallicities ([Fe/H] < -3) of two Segue 1 stars and the large metallicity spread among the members demonstrate conclusively that Segue 1 is a dwarf galaxy, and we find no evidence in favor of tidal effects. We also show that contamination by the Sagittarius stream has been overestimated. Segue 1 has the highest measured dark matter density of any known galaxy and will therefore be a prime testing ground for dark matter physics and galaxy formation on small scales.
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