Printed circuits may be electrically and mechanically connected to each other using connections such as solder connections. A first printed circuit such as a rigid printed circuit board may have solder pads and other metal traces. A second printed circuit such as a flexible printed circuit may have openings. Solder connections may be formed in the openings to attach metal traces in the flexible printed circuit to the solder pads on the rigid printed circuit board. A ring of adhesive may surround the solder connections. The flexible printed circuit may be attached to the rigid printed circuit board using the ring of adhesive. An insulating tape may cover the solder connections. A conductive shielding layer with a conductive layer and a layer of conductive adhesive may overlap the solder joints. The conductive adhesive may connect the shielding layer to the metal traces on the rigid printed circuit board.