A simple box model of the circulation into and inside the ocean cavern beneath an ice shelf is used to estimate the melt rates of Antarctic glaciers and ice shelves. The model uses simplified cavern geometries and includes a coarse parameterization of the overturning circulation and vertical mixing. The melting/freezing physics at the ice shelf/ocean interface are those usually implemented in high resolution circulation models of ice shelf caverns. The model is driven by the thermohaline inflow conditions and coupling to the heat and freshwater exchanges at the sea surface in front of the cavern. We tune the model for Pine Island Glacier and then apply it to six other major caverns. The dependence of the melting rate on thermohaline conditions at the ice shelf front is investigated for this set of caverns, including sensitivity studies, alternative parameterizations, and warming scenarios. An analytical relation between the melting rate and the inflow temperature is derived for a particular model version, showing a quadratic dependence of basal melting on a small temperature difference between the inflow and the grounding zone which changes to a linear dependence for larger differences.The model predicts melting at all ice shelf bases in agreement with observations, ranging from below a meter per year for Ronne Ice Shelf to about 25 m/y for the Pine Island Glacier. In a warming scenario with a one degree increase of the inflow temperature the latter glacier responds with a 1.4 fold increase of the melting rate. Other caverns respond by more than a ten fold increase, as e.g. Ronne Ice Shelf. The model is suitable for use as simple fast module in coarse large-scale ocean models.