We report on Westerbork 1.4 GHz radio observations of the radio counterpart to $gamma$-ray burst GRB~970508, between 0.80 and 138 days after this event. The 1.4 GHz light curve shows a transition from optically thick to thin emission between 39 and 54 days after the event. We derive the slope $p$ of the spectrum of injected electrons ($dN/dgamma_{e}proptogamma_{e}^{-p}$) in two independent ways which yield values very close to $p=2.2$. This is in agreement with a relativistic dynamically near-adiabatic blast wave model whose emission is dominated by synchrotron radiation and in which a significant fraction of the electrons cool fast.
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