To limit global warming to <2 °C we must reduce the net amount of CO2 we release into the atmosphere, either by producing less CO2 (conventional mitigation) or by capturing more CO2 (negative emissions). Here, using state-of-the-art carbon-climate models, we quantify the trade-off between these two options in RCP2.6: an Intergovernmental Panel on Climate Change scenario likely to limit global warming below 2 °C. In our best-case illustrative assumption of conventional mitigation, negative emissions of 0.5-3 Gt C (gigatonnes of carbon) per year and storage capacity of 50-250 Gt C are required. In our worst case, those requirements are 7-11 Gt C per year and 1,000-1,600 Gt C, respectively. Because these figures have not been shown to be feasible, we conclude that development of negative emission technologies should be accelerated, but also that conventional mitigation must remain a substantial part of any climate policy aiming at the 2-°C target.