In this paper, a new consistent method based on local projections for the stabilization of a Dirichlet condition is presented in the framework of finite element method with a fictitious domain approach. The presentation is made on the Poisson problem but the theoretical and numerical results can be straightforwardly extended to any elliptic boundary value problem. A numerical comparison is performed with the Barbosa-Hughes stabilization technique. The advantage of the new stabilization technique is to affect only the equation on multipliers and thus to be equation independent.