Abstract: This work addresses the problem of the numerical time-integration of nonsmooth mechanical systems subjected to unilateral contacts and impacts. The considered systems may be the standard multi-body systems or the space-discretized continuous systems obtained by using FEM approach. Up to now, two main numerical schemes are available to perform this task: the Moreau-Jean scheme which solves the constraints at the velocity level together with a Newton impact law and the Schatzman-Paoli scheme which directly considers the constraints at the position level. In both schemes, the position and velocity constraints are not both satisfied in discrete time. The aim of this work is to propose a new scheme inspired by the GGL approach in DAE that solves, in discrete time, the constraints on both position and velocity levels. The stability and the local order of the scheme will be discussed. Some comparisons with recent works on the adaption of Newmark's scheme will be presented.