Ocean melting has thinned Antarctica's ice shelves at an increasing rate over the past two decades, leading to loss of grounded ice. The Ross Ice Shelf is currently close to steady state but geological records indicate that it can disintegrate rapidly, which would accelerate grounded ice loss from catchments equivalent to 11.6 m of global sea level rise. Here, we use data from the ROSETTA-Ice airborne survey and new ocean simulations, to identify the principal threats to Ross Ice Shelf stability. We locate the tectonic boundary between East and West Antarctica from magnetic anomalies and use gravity data to generate a new high-resolution map of sub-ice-shelf bathymetry. The tectonic imprint on bathymetry constrains sub-ice-shelf ocean circulation, protecting the ice shelf grounding line from moderate changes in global ocean heat content. In contrast, local, seasonal production of warm upper-ocean water near the ice front drives rapid ice shelf melting east of Ross Island, where thinning would lead to faster grounded ice loss from both East and West Antarctic ice sheets. We confirm high modelled melt rates in this region using ROSETTA-Ice radar data. Our findings highlight the significance of both the tectonic framework and local ocean-atmosphere exchange processes near the ice front in determining the future of the Antarctic Ice Sheet.