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 \pm 0.4 ~ { } \mbox { ( stat . ) } \pm 2.0 ~ { } \mbox { ( sys . ) } \mathrm { km } ~ { } \mathrm { s } ^ { -1 } , a velocity dispersion of 0.1 ^ { +0.7 } _ { -0.1 } \mathrm { km } ~ { } \mathrm { s } ^ { -1 } , and a mean metallicity of \mbox { [ Fe / H ] } = -2.42 ^ { +0.07 } _ { -0.08 } . The upper limit on the velocity dispersion is \sigma < 1.5 \mathrm { km } ~ { } \mathrm { s } ^ { -1 } at 95.5 % confidence , and the corresponding upper limit on the mass within the half-light radius of Tuc III is 9.0 \times 10 ^ { 4 } \mathrm { M _ { \odot } } . We can not rule out mass-to-light ratios as large as 240 \mathrm { M _ { \odot } } / \mathrm { L _ { \odot } } 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 .