This paper describes a version of the Bayesian bootstrap that assigns random Dirichlet mass uniformly across statistically equivalent blocks. The method requires prior knowledge that the underlying distribution is continuous with known compact support. Under these conditions, the resulting analysis has three advantages over traditional bootstrap competitors. First, it takes account of the probability integral transformation. Secondly, it often enables exact expressions for the bootstrap distributions of common statistical functionals; thus small sample properties are known precisely, in contrast to the usual bootstrap situation. Thirdly, it shows consistent superiority in large-scale simulation experiments that compare the global accuracy of confidence intervals for the mean, median, variance and distribution function. The simulation experiments use a goodnessof-fit test as the basis for comparing the performance of competing bootstrap techniques. © 1988 Biometrika Trust.