We present the spectroscopic discovery of a broad-lined Type Ic supernova (SN 2010bh) associated with the nearby long-duration gamma-ray burst (GRB) 100316D. At z = 0.0593, this is the third-nearest GRB-SN. Nightly optical spectra obtained with the Magellan telescopes during the first week after explosion reveal the gradual emergence of very broad spectral features superposed on a blue continuum. The supernova features are typical of broad-lined SNe Ic and are generally consistent with previous supernovae associated with low-redshift GRBs. However, the inferred velocities of SN 2010bh at 21 days after explosion are a factor of ~2 times larger than those of the prototypical SN 1998bw at similar epochs, with v ~ 26,000 km/s, indicating a larger explosion energy or a different ejecta structure. A near-infrared spectrum taken 13.8 days after explosion shows no strong evidence for He I at 1.083 microns, implying that the progenitor was largely stripped of its helium envelope. The host galaxy is of low luminosity (M_R ~ -18.5 mag) and low metallicity (Z < 0.4 Z_solar), similar to the hosts of other low-redshift GRB-SNe.
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