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The land ice contribution to global mean sea level rise has not yet been predicted using ice sheet and glacier models for the latest set of socio-economic scenarios, nor using coordinated exploration of uncertainties arising from the various computer models involved. Two recent international projects generated a large suite of projections using multiple models, but primarily used previous-generation scenarios and climate models, and could not fully explore known uncertainties. Here we estimate probability distributions for these projections under the new scenarios using statistical emulation of the ice sheet and glacier models. We find that limiting global warming to 1.5 degrees Celsius would halve the land ice contribution to twenty-first-century sea level rise, relative to current emissions pledges. The median decreases from 25 to 13 centimetres sea level equivalent (SLE) by 2100, with glaciers responsible for half the sea level contribution. The projected Antarctic contribution does not show a clear response to the emissions scenario, owing to uncertainties in the competing processes of increasing ice loss and snowfall accumulation in a warming climate. However, under risk-averse (pessimistic) assumptions, Antarctic ice loss could be five times higher, increasing the median land ice contribution to 42 centimetres SLE under current policies and pledges, with the 95th percentile projection exceeding half a metre even under 1.5 degrees Celsius warming. This would severely limit the possibility of mitigating future coastal flooding. Given this large range (between 13 centimetres SLE using the main projections under 1.5 degrees Celsius warming and 42 centimetres SLE using risk-averse projections under current pledges), adaptation planning for twenty-first-century sea level rise must account for a factor-of-three uncertainty in the land ice contribution until climate policies and the Antarctic response are further constrained.