An electronic device may be provided with wireless circuitry that includes a radio-frequency transceiver circuit and an antenna. The antenna may be a patch antenna formed from a patch antenna resonating element and an antenna ground. The patch antenna resonating element may be formed from a metal patch on a printed circuit board. The antenna ground may be formed from a metal housing having a planar rear wall that lies in a plane parallel to the metal patch. The radio-frequency transceiver circuit may be coupled to the metal patch through traces on the printed circuit and may be coupled to rear wall of the housing through a screw and a screw boss in the housing. Buttons and other electrical components may be mounted on the printed circuit board and may he coupled to control circuitry on the printed circuit board through the metal patch.