We report on the results of optical follow-up observations of the counterpart of the gamma-ray burst GRB 970508, starting 7 hr after the event. Multicolor U-, B-, V-, Rc-, and Ic-band observations were obtained during the first three consecutive nights. The counterpart was monitored regularly in Rc until ~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 +/- 0.014. Deviations from a smooth power-law decay are moderate (rms = 0.15 mag). We find no flattening of the light curve at late times. The optical afterglow fluence is a significant fraction, ~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).