The Galileo spacecraft has been orbiting Jupiter since 7 December 1995, and encounters one of the four galilean satellites-Io, Europa, Ganymede and Callisto-on each orbit. Initial results from the spacecraft's magnetometer have indicated that neither Europa nor Callisto have an appreciable internal magnetic field, in contrast to Ganymede and possibly Io. Here we report perturbations of the external magnetic fields (associated with Jupiter's inner magnetosphere) in the vicinity of both Europa and Callisto. We interpret these perturbations as arising from induced magnetic fields, generated by the moons in response to the periodically varying plasma environment. Electromagnetic induction requires eddy currents to flow within the moons, and our calculations show that the most probable explanation is that there are layers of significant electrical conductivity just beneath the surfaces of both moons. We argue that these conducting layers may best be explained by the presence of salty liquid-water oceans, for which there is already indirect geological evidence in the case of Europa.