Climate sensitivity has been subjectively estimated to be likely to lie in the range of 1.5--4.5¿C, and this uncertainty contributes a substantial part of the total uncertainty in climate change projections over the coming century. Objective observationally-based estimates have so far failed to improve on this upper bound, with many estimates even suggesting a significant probability of climate sensitivity exceeding 6¿C. In this paper, we show how it is possible to greatly reduce this uncertainty by using Bayes' Theorem to combine several independent lines of evidence. Based on some conservative assumptions regarding the value of independent estimates, we conclude that climate sensitivity is very unlikely (<5% probability) to exceed 4.5¿C. We cannot assign a significant probability to climate sensitivity exceeding 6¿C without making what appear to be wholly unrealistic exaggerations about the uncertainties involved. This represents a significant lowering of the previously-estimated bound.