An electronic device may be provided that has a display. The display may produce stray light when producing images for a user. The electronic device may have an ambient light sensor for measuring ambient light levels. Ambient light data may be used in adjusting display brightness. The display may be periodically disabled to prevent the stray light from interfering with the ambient light sensor. An integrating analog-to-digital converter may be used in gathering sensor data from the ambient light sensor. Control circuitry may be configured to remove background signals from ambient light sensor data. The background signals may be associated with leakage current that arises due to offset voltages in an operational amplifier in the integrating analog-to-digital converter. The operational amplifier may have an analog autozeroing capability or control circuitry may be used to subtract background data from ambient light sensor data.