Despite substantial progress in signal source separation, results for richly structured data continue to contain perceptible artifacts. In contrast, recent deep generative models can produce authentic samples in a variety of domains that are indistinguishable from samples of the data distribution. This paper introduces a Bayesian approach to source separation that uses generative models as priors over the components of a mixture of sources, and noise-annealed Langevin dynamics to sample from the posterior distribution of sources given a mixture. This decouples the source separation problem from generative modeling, enabling us to directly use cutting-edge generative models as priors. The method achieves state-of-the-art performance for MNIST digit separation. We introduce new methodology for evaluating separation quality on richer datasets, providing quantitative evaluation of separation results on CIFAR-10. We also provide qualitative results on LSUN.