We utilise a global finite-element sea ice–ocean model (FESOM), focused on the Antarctic marginal seas, to analyse projections of ice shelf basal melting in a warmer climate. Ice shelf–ocean interaction is described using a three-equation system with a diagnostic computation of temperature and salinity at the ice–ocean interface. A tetrahedral mesh with a minimum horizontal resolution of 4 km and hybrid vertical coordinates is used. Ice shelf draft, cavity geometry, and global ocean bathymetry have been derived from the RTopo-1 data set. The model is forced with the atmospheric output from two climate models: (1) the Hadley Centre Climate Model (HadCM3) and (2) Max Planck Institute’s ECHAM5/MPI-OM coupled climate model. Results from experiments forced with their twentieth century output are used to evaluate the modelled present-day ocean state. Sea ice coverage is largely realistic in both simulations; modelled ice shelf basal melt rates compare well with observations in both cases, but are consistently smaller for ECHAM5/MPI-OM. Projections for future ice shelf basal melting are computed using atmospheric output for the Intergovernmental Panel on Climate Change (IPCC) scenarios E1 and A1B. In simulations forced with ECHAM5 data, trends in ice shelf basal melting are small. In contrast, decreasing convection along the Antarctic coast in HadCM3 scenarios leads to a decreasing salinity on the continental shelf and to intrusions of warm deep water of open ocean origin. In the case of the Filchner–Ronne Ice Shelf (FRIS), this water reaches deep into the cavity so that basal melting increases by a factor of 4 to 6 compared to the present value of about 90 Gt/year. By the middle of the twenty-second century, FRIS becomes the dominant contributor to total ice shelf basal mass loss in these simulations. Our results indicate that the surface freshwater fluxes on the continental shelves may be crucial for the future of especially the large cold water ice shelves in the Southern Ocean.