We present the first spectroscopic analysis of the faint and compact stellar system Draco II (Dra II, M_V_=-2.9+/-0.8,r_h_=19^+8^_-6_pc), recently discovered in the Panoramic Survey Telescope and Rapid Response System 1 3{PI} survey. The observations, conducted with DEIMOS on the Keck II telescope, establish some of its basic characteristics: the velocity data reveal a narrow peak with nine member stars at a systemic heliocentric velocity <v_r_>=-347.6^+1.7^_-1.8_km/s, thereby confirming Dra II is a satellite of the Milky Way; we infer a velocity dispersion with {sigma}_vr_=2.9+/-2.1km/s (<8.4km/s at the 95 per cent confidence level), which implie log_10_(M_1/2_)-5.5^+0.4^_-0.6_ and log_10_((M/L_)1/2_)=2.7^+0.5^_-0.8_, in Solar units; furthermore, very weak calcium triplet lines in the spectra of the high signal-to-noise member stars imply [Fe/H]<-2.1, whilst variations in the line strengths of two stars with similar colours and magnitudes suggest a metallicity spread in Dra II. These new data cannot clearly discriminate whether Draco II is a star cluster or amongst the faintest, most compact, and closest dwarf galaxies. However, the sum of the three - individually inconclusive - pieces of evidence presented here seems to favour the dwarf galaxy interpretation.