  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).
