Important insight into the nature of gamma-ray bursts (GRBs) has been gained in recent months mainly due to the immediate, precise localization of the bursts(1-3) and the discovery of relatively long-lived X-ray afterglows(1,4) by the satellite BeppoSAX(5). These advances have enabled deep searches which have led to the discovery of optical transients(6,7) coincident with fading X-ray sources. Optical spectroscopy of the latest burst (GRB970508; ref. 8) has clearly demonstrated that it lies at a cosmological distance, thus resolving a long-standing controversy about the distance scale to GRBs. Here we report a variable radio source within the error box of GRB970508 and coincident with the optical transient. We suggest that this is the much-sought-after radio counterpart of a GRB. If the observed fluctuations in the radio emission (’twinkling’) are a result of a strong scattering by the irregularities in the ionized Galactic interstellar gas, then the source must have an angular size of about 3 microarcseconds in the first few weeks. The damping of the fluctuations with time indicates that the source expands to a significantly larger size later on.