The computation of Bayes estimators based on mixtures of Dirichlet processes is treated. These estimators may be written as ratios of two multidimensional integrals, each of which may be decomposed into a weighted average of products of one-dimensional integrals. An importance sampling Monte Carlo method is proposed to approximate each of the weighted averages. A priori error bounds for each of the Monte Carlo estimators and a posteriori error bounds for the ratio are developed to measure the efficiency of the Monte Carlo method. Jackknife and random group error estimates are also considered. Two examples are given which illustrate the computation of the Bayes estimators.