Nonnegative matrix factorization with the minimum-volume criterion (min-vol NMF) guarantees that, under some mild and realistic conditions, the factorization has an essentially unique solution. This result has been successfully leveraged in many applications, including topic modeling, hyperspectral image unmixing, and audio source separation. In this paper, we propose a fast algorithm to solve min-vol NMF which is based on a recently introduced block majorization-minimization framework with extrapolation steps. We illustrate the effectiveness of our new algorithm compared to the state of the art on several real hyperspectral images and document data sets.