If implementation of proposals to engineer the climate through solar-radiation management (SRM) ever occurs, it is likely to be contingent on climate sensitivity. However, modelling studies examining the effectiveness of SRM as a strategy to offset anthropogenic climate change have used only the standard parameterizations of atmosphere-ocean general circulation models that yield climate sensitivities close to the Coupled Model Intercomparison Project mean. Here, we use a perturbed-physics ensemble modelling experiment to examine how the response of the climate to SRM implemented in the stratosphere (SRM-S) varies under different greenhouse-gas climate sensitivities. When SRM-S is used to compensate for rising atmospheric concentrations of greenhouse gases, its effectiveness in stabilizing regional climates diminishes with increasing climate sensitivity. However, the potential of SRM-S to slow down unmitigated climate change, even regionally, increases with climate sensitivity. On average, in variants of the model with higher sensitivity, SRM-S reduces regional rates of temperature change by more than 90% and rates of precipitation change by more than 50%.