Evidence from palaeoclimatology suggests that abrupt Northern Hemisphere cold events are linked to weakening of the Atlantic Meridional Overturning Circulation (AMOC), potentially by excess inputs of fresh water. But these insights—often derived from model runs under preindustrial conditions—may not apply to the modern era with our rapid emissions of greenhouse gases. If they do, then a weakened AMOC, as in 1975–1998, should have led to Northern Hemisphere cooling. Here we show that, instead, the AMOC minimum was a period of rapid surface warming. More generally, in the presence of greenhouse-gas heating, the AMOC’s dominant role changed from transporting surface heat northwards, warming Europe and North America, to storing heat in the deeper Atlantic, buffering surface warming for the planet as a whole. During an accelerating phase from the mid-1990s to the early 2000s, the AMOC stored about half of excess heat globally, contributing to the global-warming slowdown. By contrast, since mooring observations began in 2004, the AMOC and oceanic heat uptake have weakened. Our results, based on several independent indices, show that AMOC changes since the 1940s are best explained by multidecadal variability, rather than an anthropogenically forced trend. Leading indicators in the subpolar North Atlantic today suggest that the current AMOC decline is ending. We expect a prolonged AMOC minimum, probably lasting about two decades. If prior patterns hold, the resulting low levels of oceanic heat uptake will manifest as a period of rapid global surface warming.