Diagnosis
The mechanical thermostat has failed. It's a small device behind the control knob that signals the compressor to turn on/off.
How to Fix It
Unplug the fridge. The thermostat is usually housed in a small plastic case inside the fridge compartment, often near the light. You may need to remove a cover. Note the wire connections. Purchase the exact replacement thermostat from an appliance parts dealer using your model number. Disconnect the old one, connect the new one, reassemble.
If That Didn't Work
The problem could be the temperature sensor (if equipped) or the main control board. These are more complex diagnoses.
Heads Up
The thermostat often has a capillary tube filled with gas. Handle it carefully. Bending or kinking this tube will ruin the new part.