Government policies currently commit us to surface warming of three to four degrees Celsius above pre-industrial levels by 2100, which will lead to enhanced ice-sheet melt. Ice-sheet discharge was not explicitly included in Coupled Model Intercomparison Project phase 5, so effects on climate from this melt are not currently captured in the simulations most commonly used to inform governmental policy. Here we show, using simulations of the Greenland and Antarctic ice sheets constrained by satellite-based measurements of recent changes in ice mass, that increasing meltwater from Greenland will lead to substantial slowing of the Atlantic overturning circulation, and that meltwater from Antarctica will trap warm water below the sea surface, creating a positive feedback that increases Antarctic ice loss. In our simulations, future ice-sheet melt enhances global temperature variability and contributes up to 25 centimetres to sea level by 2100. However, uncertainties in the way in which future changes in ice dynamics are modelled remain, underlining the need for continued observations and comprehensive multi-model assessments.