(Abridged) We present densely sampled BVRI light curves of the optical transient associated with the gamma-ray burst GRB 030329, the result of a coordinated observing campaign conducted at five observatories. Augmented with published observations of this GRB, the compiled optical dataset contains 2687 photometric measurements, obtained between 78 minutes and 79 days after the burst. We show that the underlying supernova 2003dh evolved faster than, and was probably somewhat fainter than the type Ic SN 1998bw, associated with GRB 980425. We find that our data can be described by a broken power-law decay perturbed by a complex variable component. The early- and late-time decay slopes are determined to be ~1.1 and ~2, respectively. Assuming this single power-law model, we constrain the break to lie between ~3 and ~8 days after the burst. This simple, singly-broken power-law model, derived only from the analysis of our optical observations, may also account for available multi-band data, provided that the break happened ~8 days after the burst. The more complex double-jet model of Berger et al. provides a comparable fit to the optical, X-ray, mm and radio observations of this event. We detect a significant change in optical colors during the first day. Our color analysis is consistent with a cooling break frequency sweeping through the optical band during the first day. The light curves of GRB 030329 reveal a rich array of variations, superposed over the mean power-law decay. We find that the early variations are asymmetric, with a steep rise followed by a relatively slower (by a factor of about two) decline. The variations maintain a similar time scale during the first four days, and then get significantly longer.
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