We prove a Faber-Krahn inequality for the first eigenvalue of the Laplacian with Robin boundary conditions, asserting that amongst all Lipschitz domains of fixed volume, the ball has the smallest first eigenvalue. We prove the result in all space dimensions using ideas from [M.-H. Bossel, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), 47-50], where a proof for smooth domains in the plane was given. The method avoids the use of symmetrisation arguments. The results also imply variants of the Cheeger inequality for the first eigenvalue.