Open the refrigerator door to dial the built-in thermostat control to the coldest setting.
Unplug the refrigerator power cord from the wall outlet and plug in the cord to the socket on the back of the external thermostat.
Route the probe wire on the external thermostat between the refrigerator door gasket and the outer edge of the refrigerator. The probe fits between the gap in the open door and the front of the appliance. The wire is thin and flexible, and will not interfere with the normal opening and closing of the door.
Tape the probe to the interior side wall of the refrigerator as far back as possible with a piece of duct tape. Close the door.
Plug in the external thermostat's power cord to a wall outlet. This bypasses the built-in thermostat.
Dial the external thermostat's control to the desired temperature. For testing purposes, dial the unit as low as it will go. If the refrigerator starts cooling normally, the internal thermostat is defective. If the appliance still does not cool, you have a different problem, possibly involving the compressor motor assembly, the refrigerant gas condenser or a wiring problem.