We present Magellan/IMACS spectroscopy of the recently discovered Milky Way satellite Tucana III (Tuc III). We identify 26 member stars in Tuc III, from which we measure a mean radial velocity of v_hel = -102.3 +/- 0.4 (stat.) +/- 2.0 (sys.) km/s, a velocity dispersion of 0.1^+0.7_-0.1 km/s, and a mean metallicity of [Fe/H] = -2.42^+0.07_-0.08. The upper limit on the velocity dispersion is sigma < 1.5 km/s at 95.5% confidence, and the corresponding upper limit on the mass within the half-light radius of Tuc III is 9.0 x 10^4 Msun. We cannot rule out mass-to-light ratios as large as 240 Msun/Lsun for Tuc III, but much lower mass-to-light ratios that would leave the system baryon-dominated are also allowed. We measure an upper limit on the metallicity spread of the stars in Tuc III of 0.19 dex at 95.5% confidence. Tuc III has a smaller metallicity dispersion and likely a smaller velocity dispersion than any known dwarf galaxy, but a larger size and lower surface brightness than any known globular cluster. Its metallicity is also much lower than those of the clusters with similar luminosity. We therefore tentatively suggest that Tuc III is the tidally-stripped remnant of a dark matter-dominated dwarf galaxy, but additional precise velocity and metallicity measurements will be necessary for a definitive classification. If Tuc III is indeed a dwarf galaxy, it is one of the closest external galaxies to the Sun. Because of its proximity, the most luminous stars in Tuc III are quite bright, including one star at V=15.7 that is the brightest known member star of an ultra-faint satellite.
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