An analytical solution for the radiation by a prolate hemispheroidal Dielectric Resonator Antenna (DRA) over an infinite ground plane excited by a rectangular slot is presented. The dyadic Green's functions pertaining to a magnetic-current source are used in a form convenient for numerical computations. The dyadic Green's functions are then employed to formulate the electromagnetic fields radiated by the DRA. The electromagnetic far field is expressed analytically in a compact form. The far field patterns for different design parameters are computed and plotted.