In this paper, we propose a geometric integrator for nonholonomic mechanical systems. It can be applied to discrete Lagrangian systems specified through a discrete Lagrangian defined on QxQ, where Q is the configuration manifold, and a (generally nonintegrable) distribution in TQ. In the proposed method, a discretization of the constraints is not required. We show that the method preserves the discrete nonholonomic momentum map, and also that the nonholonomic constraints are preserved in average. We study in particular the case where Q has a Lie group structure and the discrete Lagrangian and/or nonholonomic constraints have various invariance properties, and show that the method is also energy-preserving in some important cases.This work has been partially supported by MEC (Spain) Grant MTM 2007-62478, project "Ingenio Mathematica" (i-MATH) No. CSD 2006-00032 (Consolider-Ingenio 2010) and Project SIMUMAT S-0505/ESP/0158 of the CAM. S. Ferraro also wants to thank SIMUMAT for a Research contract and D. Iglesias, to MEC, for a Research Contract "Juan de la Cierva".Peer reviewed