A nine-point difference scheme on a uniform mesh is considered for solving ellipitc equations with discontinuous boundary conditions in a rectangle. The convergence of the solutions of the Dirichlet difference problem is examined as a function of the disposition of the points of discontinuity of the boundary conditions relative to the mesh points. It is shown that the solution of the difference problem by the proposed scheme is uniformly convergence to the solution of the differential problem at all mesh points, if the points of discontinuity of the boundary function are located at mesh points. It is also shown that, under certain conditions, the difference scheme for Poisson's equation has second order of accuracy.