We perform an analytical and numerical study of the crossover from the Josephson effect to that of bulk superconducting flow for a one-dimensional superconductor containing a sandwich layer of normal material. A generalized Ginzburg–Landau (GL) model, proposed in Chapman & Gunzburger [1] is used in modelling the whole structure. When the thickness of the normal layer is very small, the introduction of three effective $\delta$-function potentials of specified strength leads to an exact analytical solution of the modified stationary GL equation. The resulting current density-phase offset relation is analyzed numerically. We show that the critical Josephson current density $j_c$ corresponds to a bifurcation of the solutions of the nonlinear boundary value problem for the modified GL-equation. The influence of the second term in the Fourier-decomposition of the supercurrent density-phase relation is also investigated. We derive also a simple analytical formula for the critical Josephson current.