  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.
