How emissions of black carbon (BC) aerosols affect the climate is still uncertain, due to incomplete knowledge of its sources, optical properties and atmospheric processes such as transport, removal and impact on clouds. Here we constrain simulations from four climate models with observations of atmospheric BC concentrations and absorption efficiency, and the most recent emission inventories, to show that the current global mean surface temperature change from anthropogenic BC emissions is likely to be weak at +0.03 ±0.02K. Atmospheric rapid adjustment processes are found to reduce the top of atmosphere radiative imbalance relative to instantaneous radiative forcing (direct aerosol effect) by almost 50% as a multi-model mean. Furthermore, constraining the models to reproduce observational estimates of the atmospheric vertical profile reduces BC effective radiative forcing to 0.08 W m-2, a value more than 50% lower than in unconstrained simulations. Our results imply a need to revisit commonly used climate metrics such as the global warming potential of BC. This value (for a 100-year time horizon) reduces from 680 when neglecting rapid adjustments and using an unconstrained BC profile to our best estimate of 160 ±120.