A central problem in the theory of quasiconformal and bi-Lipschitz mappings is whether they can be written as a composition of such mappings with small distortion. In this paper, we prove a decomposition result for C1 diffeomorphisms of the sphere; namely, we show that, given ε>0, every C1 diffeomorphism of the sphere Sn can be written as a composition of bi-Lipschitz mappings with isometric distortion at most 1 + ε. © 2012 London Mathematical Society.