We propose a new blind source separation algorithm based on mixtures of alpha-stable distributions. Complex symmetric alpha-stable distributions have been recently showed to better model audio signals in the time-frequency domain than classical Gaussian distributions thanks to their larger dynamic range. However, inference of these models is notoriously hard to perform because their probability density functions do not have a closed-form expression in general. Here, we introduce a novel method for estimating mixture of alpha-stable distributions based on characteristic function matching. We apply this to the blind estimation of binary masks in individual frequency bands from multichannel convolutive audio mixes. We show that the proposed method yields better separation performance than Gaussian-based binary-masking methods.