To date, few Earth System Models (ESMs) have the ability to simulate the flow in the ocean cavities below Antarctic ice shelves and its influence on basal melting. Yet capturing both this flow and the resulting melt patterns is critical for representing local, regional, and global feedbacks between the climate and sub-ice-shelf melting. Here, we present a small ensemble of historical simulations and SSP370 projections in an ESM that includes Antarctic ice-shelf cavities, the Energy Exascale Earth System Model (E3SM). The model domain has 12 km horizontal resolution around Antarctica, which is adequate for capturing dynamics in the larger ice-shelf cavities and melt fluxes aggregated across Antarctic regions. The projections show a significant increase in Antarctic melting through the 20th and 21st centuries, in contrast to the more modest, quasi-steady melt rates in the control simulation with constant 1950 conditions from which they were branched. In addition to providing an estimate of future melting under SSP370, these simulations are also a stepping stone to coupled ice sheet-ocean simulations planned for the near future. The coupling is described in the companion presentation “Antarctic Ice-Ocean Interactions in the Energy Exascale Earth System Model.”