After an engine automatic stop processing is started, a diagnosis of a starter is started at a time of t1 when an engine speed drops to a diagnosis starting speed. Before the engine speed drops to the lower limit speed at which an engine can be restarted without being cranked by the starter, the diagnosis of the starter is conducted. In the diagnosis of the starter, it is determined whether the starter is faulty based on a current passing through the starter when a relay is turned off to stop energization of the starter before the starter actually rotates after the relay is turned on. Thereby, a malfunction of the starter can be detected before the engine is automatically stopped. It is avoided to automatically stop the engine which is incapable of being automatically restarted due to the malfunction of the starter.