We present early observations of the afterglow of GRB 030329 and thespectroscopic discovery of its associated supernova SN 2003dh. Weobtained spectra of the afterglow of GRB 030329 each night from March30.12 (0.6 days after the burst) to April 8.13 (UT) (9.6 days after theburst). The spectra cover a wavelength range of 350-850 nm. The earlyspectra consist of a power-law continuum(Fν~ν-0.9) with narrow emission linesoriginating from H II regions in the host galaxy, indicating a lowredshift of z=0.1687. However, our spectra taken after 2003 April 5 showbroad peaks in flux characteristic of a supernova. Correcting for theafterglow emission, we find that the spectrum of the supernova isremarkably similar to the Type Ic ``hypernova'' SN 1998bw. While thepresence of supernovae has been inferred from the light curves andcolors of gamma-ray burst afterglows in the past, this is the firstdirect, spectroscopic confirmation that a subset of classical gamma-raybursts originate from supernovae.Based on data from the Multiple Mirror Telescope Observatory 6.5 mtelescope, the Magellan 6.5 m Clay telescope, and the Fred LawrenceWhipple Observatory 1.5 m telescope.