The recent detection of delayed X-ray and optical emission--``afterglow''--associated with γ-ray bursts (GRBs) supports models in which the bursts are produced by relativistic expanding blast waves--``fireballs''--at cosmological distances. The detection of absorption lines in the optical afterglow of the GRB of 1997 May 8 confirms that the sources lie at cosmological distance. We show here that the new features detected in GRB 970508 afterglow, radio emission 1 week following the burst and a 2 day increase in optical flux, are consistent with the blast wave model. The fireball optical depth at radio frequencies is much smaller than previously estimated, which accounts for the observed radio emission. The initial suppression of optical flux is consistent with that predicted from electron cooling. The combined radio and optical data imply that the fireball energy is ~1052 ergs and that the density of the medium into which the blast wave expands is ~1 cm-3, a value typical for gas within galaxies. We predict the time dependence of the radio flux and the absorption frequency, which constitute tests of the fireball model as described in this Letter.