The release of carbon from tropical forests may exacerbate future climate change(1), but the magnitude of the effect in climate models remains uncertain(2). Coupled climate-carbon-cycle models generally agree that carbon storage on land will increase as a result of the simultaneous enhancement of plant photosynthesis and water use efficiency under higher atmospheric CO2 concentrations, but will decrease owing to higher soil and plant respiration rates associated with warming temperatures(3). At present, the balance between these effects varies markedly among coupled climate-carbon-cycle models, leading to a range of 330 gigatonnes in the projected change in the amount of carbon stored on tropical land by 2100. Explanations for this large uncertainty include differences in the predicted change in rainfall in Amazonia(4,5) and variations in the responses of alternative vegetation models to warming(6). Here we identify an emergent linear relationship, across an ensemble of models(7), between the sensitivity of tropical land carbon storage to warming and the sensitivity of the annual growth rate of atmospheric CO2 to tropical temperature anomalies(8). Combined with contemporary observations of atmospheric CO2 concentration and tropical temperature, this relationship provides a tight constraint on the sensitivity of tropical land carbon to climate change. We estimate that over tropical land from latitude 30 degrees north to 30 degrees south, warming alone will release 53 +/- 17 gigatonnes of carbon per kelvin. Compared with the unconstrained ensemble of climate-carbon-cycle projections, this indicates a much lower risk of Amazon forest die-back under CO2-induced climate change if CO2 fertilization effects are as large as suggested by current models(9). Our study, however, also implies greater certainty that carbon will be lost from tropical land if warming arises from reductions in aerosols(10) or increases in other greenhouse gases(11).