Mixed norms that promote structured sparsity have numerous applications in signal processing and machine learning problems. In this work we present a new algorithm, based on a Newton root search technique, for computing the projection onto the ℓ_∞,1 ball, which has found application in cognitive neuroscience and classification tasks. Numerical simulations show that our proposed method is between 8 and 10 times faster on average, and of up to 20 times faster for very sparse solutions, than the previous state of the art. Tests on real functional magnetic resonance image data show that, for some data distributions, our algorithm can obtain speed improvements by a factor of more than 100.