This paper is concerned with the structure of the genealogy of a sample in which it is observed that some subset of chromosomes carries a particular mutation, assumed to have arisen uniquely in the history of the population. A rigorous theoretical study of this conditional genealogy is given using coalescent methods. Particular results include the mean, variance, and density of the age of the mutation conditional on its frequency in the sample. Most of the development relates to populations of constant size, but we discuss the extension to populations which have grown exponentially to their present size.