The electromagnetic field generated by a vertical electric dipole in the air over the surface of a two-layered region is determined for continuous-wave excitation. The region of interest consists of a conductor or dielectric with high permittivity, coated with an electrically thin layer of a dielectric under a half-space of air. Simple explicit formulas are derived for the field at all points in all three regions including the surface wave. Typical applications are to microstrip circuits and antennas, communication over the earth when this is coated with a layer of asphalt or cement, and communication over the sea or a lake when this is under a layer of ice.