The sea-level contribution of the Antarctic ice sheet constitutes a large uncertainty in future sea-level projections. Here we apply a linear response theory approach to 16 state-of-the-art ice sheet models to estimate the Antarctic ice sheet contribution from basal ice-shelf melting within the 21st century. In assuming linear response theory, we are able to capture complex temporal responses of the ice sheets but neglect any self-dampening or self-amplifying processes. We include uncertainty in the atmospheric warming response to carbon emissions, uncertainty in the oceanic transport to the Southern Ocean, and the observed range of responses of basal ice-shelf melting to oceanic warming outside the ice shelf cavity. Uncertainty in basal ice shelf melting is convoluted with the linear response functions of each of the 16 ice sheet models to obtain the ice flow response to the individual global warming path. The model median for the observational period from 1992 to 2017 of the ice loss due to basal ice shelf melting is 10.2 mm, with a likely range between 5.2 and 21.3 mm (compared to 7.4 mm of observed global sea-level rise, with a standard deviation of 3.7 mm over this same time period). For RCP8.5, we obtain a median contribution of the Antarctic ice sheet to global mean sea level rise from basal ice-shelf melting within the 21st century of 17 cm, with a likely range between 9 and 36 cm and a very likely range between 6 and 58 cm. For RCP2.6, the procedure yields a median of 13 cm of global mean sea-level contribution. The likely range for the RCP2.6 scenario is between 7 and 24 cm, and the very likely range is between 4 and 37 cm. We also provide projections for five Antarctic regions and for each model and each emissions scenario separately. The rate of sea-level contribution is highest under the RCP8.5 scenario with a 21st century maximum of 4 cm per decade (median value) and a likely range between 2 and 9 cm per decade.