  The recent detection of delayed X-ray and optical emission, ``afterglow,''
associated with gamma-ray bursts (GRBs) supports models, where the bursts are
produced by relativistic expanding blastwaves, ``fireballs,'' at cosmological
distances. The detection of absorption lines in the optical afterglow of the
GRB of 8 May 1997 confirms that the sources lie at cosmological distance. We
show here that the new features detected in GRB970508 afterglow, radio emission
one week following the burst and a 2 day increase in optical flux, are
consistent with the blastwave 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 due to electron cooling. The combined radio and optical
data imply that the fireball energy is $\sim10^{52}$erg, and that the density
of the medium into which the blastwave expands is $\sim 1cm^{-3}$, 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 paper.
