In this article 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 \textit{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.