We report on the results of optical follow-up observations of the counterpart of GRB 970508, starting 7 hours after the event. Multi-color U, B, V, R$_{c}$ and I$_{c}$ band observations were obtained during the first three consecutive nights. The counterpart was monitored regularly in R$_{c}$ until $sim$ 4 months after the burst. The light curve after the maximum follows a decline that can be fitted with a power law with exponent $alpha$ = --1.141 $pm$ 0.014. Deviations from a smooth power law decay are moderate (r.m.s. = 0.15 magnitude). We find no flattening of the light curve at late times. The optical afterglow fluence is a significant fraction, $sim$ 5%, of the GRB fluence. The optical energy distribution can be well represented by a power law, the slope of which changed at the time of the maximum (the spectrum became redder).
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