An oblate semispheroidal cavity with metallic walls flush-mounted under a metallic ground plane is coupled to the half-space above the ground plane through an aperture that corresponds to the interfocal disk of the oblate spheroidal system. A diaphragm is located across the aperture. The diaphragm material, the material filling the cavity, and the half-space are isorefractive to each other. A new exact solution is obtained for the radiation of an electric or a magnetic dipole located on the symmetry axis of the structure and axially oriented. The exact solution is expressed in terms of series containing oblate spheroidal functions. These series are evaluated to provide a benchmark solution for a primary source located either inside the cavity below the diaphragm or outside the cavity in the unbounded medium.