Before NASA’s Dawn mission, the dwarf planet Ceres was widely believed to contain a substantial ice-rich layer below its rocky surface. The existence of such a layer has significant implications for Ceres’s formation, evolution, and astrobiological potential. Ceres is warmer than icy worlds in the outer Solar System and, if its shallow subsurface is ice-rich, large impact craters are expected to be erased by viscous flow on short geologic timescales. Here we use digital terrain models derived from Dawn Framing Camera images to show that most of Ceres’s largest craters are several kilometres deep, and are therefore inconsistent with the existence of an ice-rich subsurface. We further show from numerical simulations that the absence of viscous relaxation over billion-year timescales implies a subsurface viscosity that is at least one thousand times greater than that of pure water ice. We conclude that Ceres’s shallow subsurface is no more than 30% to 40% ice by volume, with a mixture of rock, salts and/or clathrates accounting for the other 60% to 70%. However, several anomalously shallow craters are consistent with limited viscous relaxation and may indicate spatial variations in subsurface ice content.