Although more than 2,000 astronomical gamma-ray bursts (GRBs) have been detected, and numerous models proposed to explain their occurrence(1), they have remained enigmatic owing to the lack of an obvious counterpart at other wavelengths(2-5). The recent groundbased detection(6,7) of a transient optical source in the vicinity of GRB970228 (refs 8-11) may therefore have provided a breakthrough. The optical counterpart appears to be embedded in an extended source which, if a galaxy as has been suggested(7,12), would lend weight to those models that place GRBs at cosmological distances. Here we report observations using the Hubble Space Telescope of the transient counterpart and extended source 26 and 39 days after the initial gamma-ray outburst. We find that the counterpart has faded since the initial detection (and continues to fade), but the extended source exhibits no significant change in brightness between the two dates of the observations reported here, The size and apparent constancy of the extended source imply that it is extragalactic, but its faintness makes a definitive statement about its nature difficult. Nevertheless, the decay profile of the transient source is consistent with a popular impulsive-fireball model(13), which assumes a merger between two neutron stars in a distant galaxy.