We report on Westerbork 1.4 GHz radio observations of the radio counterpart
to γ-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/dγe∝γe−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.Comment: Paper I. Accepted for publication in the Astrophysical Journal
Letter