Control apparatus for automatically stopping and restarting an engine

A control apparatus for automatically stopping and restarting an engine of a vehicle is provided, which is equipped with an engine, a motor, a compressor and an air conditioner. The apparatus has a first section for making a judgment on stopping the engine and a second section for making a judgment on restarting the engine. The apparatus has features that the apparatus provides the air conditioner with a plurality of operational modes selected by a user, and when the vehicle is not in motion with selection of a first mode, the first section permits the engine to stop if a first power that the motor can supply is greater than a second power of the compressor required by the air conditioner, and the second section permits the engine to restart if the second power exceeds the first power.