Ground-based spectroscopy of Jupiter's moon Europa, combined with gravity data, suggests that the satellite has an icy crust roughly 150 km thick and a rocky interior. In addition, images obtained by the Voyager spacecraft revealed that Europa's surface is crossed by numerous intersecting ridges and dark bands (called lineae) and is sparsely cratered, indicating that the terrain is probably significantly younger than that of Ganymede and Callisto. It has been suggested that Europa's thin outer ice shell might be separated from the moon's silicate interior by a liquid water layer, delayed or prevented from freezing by tidal heating; in this model, the lineae could be explained by repetitive tidal deformation of the outer ice shell. However, observational confirmation of a subsurface ocean was largely frustrated by the low resolution (>2 km per pixel) of the Voyager images. Here we present high-resolution (54 m per pixel) Galileo spacecraft images of Europa, in which we find evidence for mobile 'icebergs'. The detailed morphology of the terrain strongly supports the presence of liquid water at shallow depths below the surface, either today or at some time in the past. Moreover, lower-resolution observations of much larger regions suggest that the phenomena reported here are widespread.