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 \pm 1.3 km s ^ { -1 } . We measure an internal velocity dispersion of 4.3 \pm 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 } \times 10 ^ { 5 } M _ { \odot } 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 . Although Segue 1 spatially overlaps the leading arm of the Sagittarius stream , its velocity is 100 km s ^ { -1 } different than that predicted for recent Sagittarius tidal debris at this position . We can not rule out the possibility that Segue 1 has been tidally disrupted , but 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 \pm 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 .