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 3π 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_rrangle =-347.6^{+1.7}_{-1.8} km s^{-1}, thereby confirming Dra II is a satellite of the Milky Way; we infer a velocity dispersion with σ_vr = 2.9 ± 2.1 km s^-1 (<8.4 km s^-1 at the 95 per cent confidence level), which implies 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.