The blind source separation problem is to extract the underlying source signals from a set of linear mixtures, where the mixing matrix is unknown. We consider a two-stage separation process. First, a priori selection of a possibly overcomplete signal dictionary (e.g. wavelet frame, learned dictionary, etc.) in which the sources are assumed to be sparsely representable. Second, unmixing the sources by exploiting the their sparse representability. We consider the general case of more sources than mixtures. But also derive a more efficient algorithm in the case of a non-overcomplete dictionary and equal numbers of sources and mixtures. Experiments with artificial signals and with musical sounds demonstrate significantly better separation than other known techniques.