The continued increase in the atmospheric concentration of carbon dioxide due to anthropogenic emissions is predicted to lead to significant changes in climate. About half of the current emissions are being absorbed by the ocean and by land ecosystems, but this absorption is sensitive to climate as well as to atmospheric carbon dioxide concentrations, creating a feedback loop. General circulation models have generally excluded the feedback between climate and the biosphere, using static vegetation distributions and CO2 concentrations from simple carbon-cycle models that do not include climate change. Here we present results from a fully coupled, three-dimensional carbon-climate model, indicating that carbon-cycle feedbacks could significantly accelerate climate change over the twenty-first century. We find that under a `business as usual' scenario, the terrestrial biosphere acts as an overall carbon sink until about 2050, but turns into a source thereafter. By 2100, the ocean uptake rate of 5GtCyr-1 is balanced by the terrestrial carbon source, and atmospheric CO2 concentrations are 250p.p.m.v. higher in our fully coupled simulation than in uncoupled carbon models, resulting in a global-mean warming of 5.5K, as compared to 4K without the carbon-cycle feedback.