Cumulative emissions drive peak global warming and determine the safe carbon budget compatible with staying below 2oC or 1.5oC. The safe carbon budget is lower if uncertainty about the transient climate response is high and risk tolerance low. Together with energy costs this budget determines the constrained welfare-maximizing carbon price and how quickly fossil fuel is replaced by renewable energy and how much of it is abated. This price is the sum of a gradual damages component familiar from the unconstrained optimal carbon price highlighted in economic studies and a Hotelling component for the additional price needed to ensure that the safe carbon budget is never violated familiar from IAM studies. If policy makers ignore damages, as in the cost-minimizing temperature constraint literature, a more rapidly rising carbon price results. The alternative of adjusting damages upwards to factor in the peak warming constraint leads initially to a higher carbon price which rises less rapidly.