Several lines of evidence suggest that Saturn's largest moon, Titan, has a global subsurface ocean beneath an outer ice shell 50 to 200 kilometres thick. If convection is occurring, the rigid portion of the shell is expected to be thin; similarly, a weak, isostatically compensated shell has been proposed to explain the observed topography. Here we report a strong inverse correlation between gravity and topography at long wavelengths that are not dominated by tides and rotation. We argue that negative gravity anomalies (mass deficits) produced by crustal thickening at the base of the ice shell overwhelm positive gravity anomalies (mass excesses) produced by the small surface topography, giving rise to this inverse correlation. We show that this situation requires a substantially rigid ice shell with an elastic thickness exceeding 40 kilometres, and hundreds of metres of surface erosion and deposition, consistent with recent estimates from local features. Our results are therefore not compatible with a geologically active, low-rigidity ice shell. After extrapolating to wavelengths that are controlled by tides and rotation, we suggest that Titan's moment of inertia may be even higher (that is, Titan may be even less centrally condensed) than is currently thought.