We have reconstructed the spectrum of the afterglow of GRB 970508, on May 21.0 UT (12.1 days after the GRB), on the basis of observations spanning the X-ray to radio range. The low-frequency power law index of the spectrum, $alpha=0.44pm0.07$ ($F_ upropto u^alpha$), is in agreement with the expected value $alpha=1/3$ for optically thin synchrotron radiation. The 1.4 GHz emission is self-absorbed. We infer constraints on the break frequecies $ u_{c}$ and $ u_{m}$ on May 21.0 UT from a spectral transition from $F_{ u} sim u^{-0.6}$ to $F_{ u} sim u^{-1.1}$ in the optical passband around 1.4 days. A model of an adiabatically expanding blast wave, emitting synchrotron radiation, in which a significant fraction of the electrons cool rapidly provides a successful and consistent description of the afterglow observations over nine decades in frequency, ranging in time from trigger until several months later.
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