In this talk we consider a model first order mean field game problem, introduced by J.M. Lasry and P.L. Lions. Its solution (v,m) can be obtained as the limit of the solutions of the second order mean field game problems, when the noise parameter tends to zero. We propose a semi-discrete in time approximation of the system and, under natural assumptions, we prove that it is well posed and that it converges to (v,m) when the discretization parameter tends to zero.