We demonstrate that in a mesoscopic, fully phase coherent, superconductor--two-dimensional-electron-gas--superconductor structure the period of the critical current versus flux through the junction is $h/e$, i.e., twice the superconducting flux quantum. Using a heuristic approach we argue that the observations point to a nonlocal supercurrent density. The particles carrying the supercurrent sense the superconductor phase, which is spatially modulated by the magnetic field, along the entire electrode.