Electronic devices may be provided that contain wireless communications circuitry. The wireless communications circuitry may include antenna structures that are formed from an internal ground plane and a peripheral conductive housing member. A conductive path may be formed that connects the peripheral conductive housing member and the internal ground plane. The conductive path may include a flex circuit. The flex circuit can include a solder flow barrier to constrain flow of solder. A metal structure may be welded to the peripheral conductive housing member. A solder pad and other traces in the flex circuit may be soldered to the metal structure at one end of the conductive path. At the other end of the conductive path, the flex circuit may be attached to the ground plane using a bracket, screw, and screw boss.