A stratified scavenging two-cycle engine includes: an air passage for delivering pure air to a scavenging passage; an air valve for opening and closing the air passage; and an auxiliary air passage for delivering the pure air to the scavenging passage when the air valve is completely closed. An amount of air for air-fuel mixture sucked in a crank chamber is reduced to increase the concentration of the air-fuel mixture, and air that supplements the reduced amount of the air for air-fuel mixture is fed into the scavenging passage through the auxiliary air passage, which enables the engine to be smoothly accelerated even when being suddenly accelerated from an idling state with an appropriate concentration of the air-fuel mixture sucked in a cylinder chamber during idling.