Climate adaptation and flood risk assessments(1,2) have incorporated sea-level rise (SLR) projections developed using semi-empirical methods(3-5) (SEMs) and expert-informed mass-balance scenarios(2,6). These techniques, which do not explicitly model ice dynamics, generate upper bounds on twenty-first century SLR that are up to three times higher than Intergovernmental Panel on Climate Change estimates(7). However, the physical basis underlying these projections, and their likelihood of occurrence, remain unclear(8-10). Here, we develop mass-balance projections for the Antarctic ice sheet within a Bayesian probabilistic framework(10), integrating numerical model output(11) and updating projections with an observational synthesis(12). Without abrupt, sustained, changes in ice discharge (collapse), we project a 95th percentile mass loss equivalent to similar to 13 cm SLR by 2100, lower than previous upper-bound projections. Substantially higher mass loss requires regional collapse, invoking dynamics that are likely to be inconsistent with the underlying assumptions of SEMs. In this probabilistic framework, the pronounced sensitivity of upper-bound SLR projections to the poorly known likelihood of collapse is lessened with constraints on the persistence and magnitude of subsequent discharge. More realistic, fully probabilistic, estimates of the ice-sheet contribution to SLR may thus be obtained by assimilating additional observations and numerical models(11,13).