A control device performs fuel injection without cranking when engine-speed Ne is within a first-range (Ne>N1) and an engine-restart request occurs. The device performs the cranking after the synchronization in rotation-speed between pinion and ring gears, and the gear meshing operation between them when the engine-speed Ne is within a second range (N1≧Ne>N2) and the engine-restart request occurs. The device performs the cranking after the gear meshing operation and drives a starter to rotate the pinion gear when the engine-speed Ne is within a third-range (N3≧Ne) and the above request occurs. When the above request occurs in an engine restart ready-range (N2≧Ne>N3), the control device performs the cranking after the engine-speed Ne is decreased to a value within the third-range (N3≧Ne) and after the completion of the gear meshing operation.