  We present Keck/DEIMOS spectroscopy of Segue 1, an ultra-low luminosity (M_V
= -1.5) Milky Way satellite companion. While the combined size and luminosity
of Segue 1 are consistent with either a globular cluster or a dwarf galaxy, we
present spectroscopic evidence that this object is a dark matter-dominated
dwarf galaxy. We identify 24 stars as members of Segue 1 with a mean
heliocentric recession velocity of 206 +/- 1.3 kms. We measure an internal
velocity dispersion of 4.3+/-1.2 kms. Under the assumption that these stars are
in dynamical equilibrium, we infer a total mass of 4.5^{+4.7}_{-2.5} x 10^5
Msun in the case where mass-follow-light; using a two-component maximum
likelihood model, we determine a similar mass within the stellar radius of 50
pc. This implies a mass-to-light ratio of ln(M/L_V) = 7.2^{+1.1}_{-1.2} or
M/L_V = 1320^{+2680}_{-940}. The error distribution of the mass-to-light ratio
is nearly log-normal, thus Segue 1 is dark matter-dominated at a high
significance. Using spectral synthesis modeling, we derive a metallicity for
the single red giant branch star in our sample of [Fe/H]=-3.3 +/- 0.2 dex.
Finally, we discuss the prospects for detecting gamma-rays from annihilation of
dark matter particles and show that Segue 1 is the most promising satellite for
indirect dark matter detection. We conclude that Segue 1 is the least luminous
of the ultra-faint galaxies recently discovered around the Milky Way, and is
thus the least luminous known galaxy.
