1. An automatic stop and start control system for a multiple-cylinder internal combustion engine mounted on a vehicle, said control system controls automatically stopping and restarting said engine according to operating conditions of said vehicle and said engine, said automatic stop and start control system comprising: fuel supply stop control means for providing a fuel supply to at least one of the cylinders of said engine, when an engine stop condition for stopping said engine is satisfied, and subsequently stopping the fuel supply to all of the cylinders of said engine; and fuel supply start control means for immediately providing the fuel supply to at least one of said cylinders at the time an engine restart condition for restarting said engine is satisfied.