An electronic device may be provided with a display cover layer mounted to the device using an adhesive bond with a display. The display may be a flexible display. The flexible display may include Organic Light Emitting Diode display technology. The display may be mounted to a rigid support structure. The rigid support structure may be mounted to a device housing member. Mounting the display cover layer to the display may eliminate the need to mount the display cover layer to the device housing and may allow active display pixels to be visible under the display cover layer closer to the device housing than in conventional devices. Providing the electronic device with active display pixels closer to the device housing may reduce the need for an inactive border around the display and may improve the aesthetic appeal of the electronic device.