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We present Keck DEIMOS spectroscopy of stars in eight of the newly discovered ultra-faint dwarf galaxies around the Milky Way. We measure the velocity dispersions of Canes Venatici I, Canes Venatici II, Coma Berenices, Hercules, Leo IV, Leo T, Ursa Major I, and Ursa Major II from the velocities of 18-214 stars in each galaxy and find dispersions ranging from 3.3 to 7.6 km s^(-1). The six galaxies with absolute magnitudes M_V < -4 are highly dark matter dominated, with mass-to-light ratios approaching 1000 M_☉/L_(☉,V). For the fainter galaxies we find tentative evidence for tidal disruption. The measured velocity dispersions of the ultra-faint dwarfs are correlated with their luminosities, indicating that a minimum mass for luminous galactic systems may not yet have been reached. We also measure the metallicities of the observed stars and find that the new dwarfs have mean metallicities of [Fe/H] = -2.0 to -2.3; these galaxies represent some of the most metal-poor stellar systems known. The six brightest of the ultra-faint dwarfs extend the luminosity-metallicity relationship followed by more luminous dwarfs by a factor of ~30 in luminosity. We detect metallicity spreads of up to 0.5 dex in several objects, suggesting multiple star formation epochs. UMa II and Com, despite their exceptionally low luminosities, have higher metallicities that suggest they may once have been much more massive. Having established the masses of the ultra-faint dwarfs, we re-examine the missing satellite problem. After correcting for the sky coverage of the Sloan Digital Sky Survey, we find that the ultra-faint dwarfs substantially alleviate the discrepancy between the predicted and observed numbers of satellites around the Milky Way, but there are still a factor of ~4 too few dwarf galaxies over a significant range of masses. We show that if galaxy formation in low-mass dark matter halos is strongly suppressed after reionization, the simulated circular velocity function of CDM subhalos can be brought into approximate agreement with the observed circular velocity function of Milky Way satellite galaxies.