The purpose of this paper is to provide a priori error estimates on the approximation of contact conditions in the framework of the eXtended Finite-Element Method (XFEM). This method allows to perform nite-element computations on cracked domains by using meshes of the non-cracked domain. We consider a stabilized Lagrange multiplier method whose particularity is that no discrete inf-sup condition is needed in the convergence analysis. The contact condition is prescribed on the crack with a discrete multiplier which is the trace on the crack of a nite-element method on the non-cracked domain, avoiding the de nition of a speci c mesh of the crack. Additionally, we present numerical experiments which con rm the e ciency of the proposed method