We present a new approach to estimate the magnitude of global-mean cooling (dTLGM) at the Last Glacial Maximum (LGM) relative to the pre-industrial climate, by combining an ensemble of coupled climate model simulations with empirical constraints on regional cooling inferred from proxy data. We have generated a large ensemble of paired runs (~100) for pre-industrial and LGM boundary conditions with different versions of the same climate model of intermediate complexity. The model ensemble covers a broad range of climate sensitivities and produces a similarly broad range of dTLGM (4.3--9.8¿C). Using reconstructed tropical SST cooling, we constrain the range of dTLGM to 5.8 ¿ 1.4¿C, which is corroborated by proxy data from other regions. This cooling is considerably larger than most estimates of previous LGM simulations. The reason is that most models did not account for the effect of atmospheric dust content and vegetation changes, which yield an additional 1.0--1.7¿C global cooling.