The fossil record provides a lower bound on the primate divergence time of 54.8 million years ago, but does not provide an explicit estimate for the divergence time itself. We show how the pattern of diversification through the Cenozoic can be combined with a model for speciation to give a distribution for the age of the primates. The primate fossil record, the number of extant primate species, and information about the structure of the primate phylogenetic tree are combined to provide an estimate for the joint distribution of the primate and anthropoid divergence times. To take this information into account, we derive the structure of the birth-and-death process conditioned to have a subtree originate at a particular point in time. This process has a size-biased law and has an immortal line running from the root of the tree to the root of the subtree, with species on the spine having modified offspring and length distributions. We conclude that it is not possible, with this model, to rule out a Cretaceous origin for the primates.