We present Keck/DEIMOS spectroscopy of Segue 1, an ultra-low-luminosity (M_V = –1.5^+0.6 _–0.8) 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  km s^–1. Although Segue 1 spatially overlaps the leading arm of the Sagittarius stream, its velocity is 100 km s^–1 different from that predicted for recent Sagittarius tidal debris at this position. We measure an internal velocity dispersion of 4.3 ± 1.2  km s^–1. Under the assumption that these stars are gravitationally bound and in dynamical equilibrium, we infer a total mass of 4.5^+4.7 _–2.5 × 10⁵ M _☉ in the mass-follow-light case; using a two-component maximum-likelihood model, we determine a mass within 50 pc of 8.7⁺¹³ _–5.2 × 10⁵ M _☉. These imply mass-to-light (M/L) ratios of ln(M/L _V ) = 7.2^+1.1 _–1.2 (M/L _V = 1320⁺²⁶⁸⁰ _–940) and M/L _V = 2440⁺¹⁵⁸⁰ _–1775, respectively. The error distribution of the M/L is nearly lognormal, thus Segue 1 is dark matter-dominated at a high significance. Although we cannot rule out the possibility that Segue 1 has been tidally disrupted, we do not find kinematic evidence supporting tidal effects. 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.