  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 ($dN/d\gamma_{e}\propto\gamma_{e}^{-p}$) 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.
