We study a Finite Volume discretization of a strongly coupled elliptic-parabolic PDE system describing miscible displacement in a porous medium. We discretize each equation by a finite volume scheme whose properties are to handle a wide variety of unstructured grids (in any space dimension) and to give strong enough convergence to handle the nonlinear coupling of the equations. We prove the convergence of the scheme as the time and space steps go to $0$. Finally, we provide numerical results which show that, from a practical point of view, the scheme behaves very well.