We report the fabrication and measurement of Corbino geometry superconductor-normal metal-superconductor Josephson junctions. A circular junction barrier is defined in a superconductor-normal metal bilayer and bias current flows through the device in the radial direction. In contrast to conventional junction geometries, the junction barrier is entirely surrounded by a superconducting loop: this implies that flux can enter the junction only as single quanta. We have observed abrupt suppression/reappearance of critical current corresponding to flux entry/annihilation events as the external magnetic field is varied. Furthermore, when the width of the superconducting film enclosing the junction is sufficiently small we observe critical current suppression/reappearance in a series of jumps, suggesting incomplete quantization of magnetic flux in a superconducting film over mesoscopic length scales. We present a theoretical simulation of device response to the approach of a single vortex. A consideration of the actual device geometry shows that screening currents in the edge of the film when an external magnetic field is applied to a bilayer strip containing the junction create an effective double-well potential for confined flux vortices trapped in the junction. Further experiments and possible applications of such devices are discussed.