Solar radiation management (SRM) geoengineering has been proposed as one means of helping avoid the occurrence of dangerous climate change and undesirable state transitions ('tipping points') in the Earth system. The irreversible melting of the Greenland Ice Sheet is a case in point—a state transition that could occur as a result of CO₂-driven elevated global temperatures, and one leading to potentially catastrophic sea-level rise. SRM schemes such as the creation of a 'sunshade' or injection of sulfate aerosols into the stratosphere could reduce incoming solar radiation, and in theory balance, in a global mean, the greenhouse warming resulting from elevated concentrations of CO₂ in the atmosphere. Previous work has highlighted that a geoengineered world would have: warming towards the poles, cooling in the tropics, and a reduction in the global hydrological cycle, which may have important implications for the Greenland Ice Sheet. Using a fully coupled global climate model in conjunction with an ice sheet model, we assess the consequences for the mass balance of the Greenland Ice Sheet of the reorganization of climate patterns by the combination of high CO₂ and geoengineering. We find that Greenland surface temperature and precipitation anomalies, compared to the pre-industrial situation, decrease almost linearly with increasing levels of SRM geoengineering, but that these combine to create a highly non-linear response of the ice sheet. The substantial melting of the Greenland Ice Sheet predicted for four times pre-industrial CO₂ levels is prevented in our model with only a partial application of SRM, and hence without having to fully restore the global average temperature back to pre-industrial levels. This suggests that the degree of SRM geoengineering required to mitigate the worst impacts of greenhouse warming, such as sea-level rise, need not be as extensive as generally assumed.