We present extensive optical and infrared photometry of the afterglow of gamma-ray burst (GRB) 030329 and its associated supernova (SN) 2003dh over the first two months after detection (2003 March 30-May 29 UT). Optical spectroscopy from a variety of telescopes is shown and, when combined with the photometry, allows an unambiguous separation between the afterglow and SN contributions. The optical afterglow of the GRB is initially a power-law continuum but shows significant color variations during the first week that are unrelated to the presence of an SN. The early afterglow light curve also shows deviations from the typical power-law decay. An SN spectrum is first detectable åisebox-0.5ex 7 days after the burst and dominates the light after i̊sebox-0.5ex 11 days. The spectral evolution and the light curve are shown to closely resemble those of SN 1998bw, a peculiar Type Ic SN associated with GRB 980425, and the time of the SN explosion is close to the observed time of the GRB. It is now clear that at least some GRBs arise from core-collapse SNe.