We propose a finite volume scheme to discretize the one-dimensional Vlasov-Po- isson system, we prove that, if the initial data is positive, bounded, continuous, and has its first moment bounded, then the numerical approximation converges to the weak solution of the system for the weak topology of $L^\infty$. Moreover, if the initial data belongs to $BV$, the convergence is strong in $C^0(0,T;L^1_loc)$. To prove the convergence of the discrete electric field, we obtain an estimation in $W^1,\infty(\Omega_T)$.}