Projections of future transient global temperature increase in climate models for a known forcing depend on the strength of the atmospheric feedbacks and the rate of transient ocean heat uptake. A Bayesian framework and an intermediate complexity climate model are used to calculate a probability density function (PDF) of the transient climate response (TCR), constrained by observations of global surface warming and ocean heat uptake. The PDF constrained by observations is wider than the TCR range of current climate models, and has a slightly lower mean. Uncertainties in the observed ocean warming are shown to potentially affect the TCR. It is proposed, however, that even if models were found to overestimate ocean heat uptake, correcting that bias would lead to revisions in surface temperature projections over the twenty-first century that are smaller than the uncertainties introduced by poorly quantified atmospheric feedbacks.