An air conditioning system for an automobile having an evaporator, in a refrigerating circuit, and a heater core which is located downstream from the evaporator in a duct having an inlet for introduction of air and an outlet for discharging the air into a cabin, and an air mix damper for controlling the ratio of the amount of air into the heater core and the amount of air by-passing the heater core. The degree of the opening of the air mix damper is controlled in accordance with the cooling capacity at the evaporator sensed by an after-evaporator temperature sensor so that a target value of the temperature of the air discharged into the cabin is obtained. A correction of the degree of opening of the air mix damper is done by estimating an actual temperature at the after-evaporator sensor during a transient state of the system between the ON and OFF conditions of a compressor, so that a desired setting of the air conditioning state is maintained. The target air temperature is not corrected if it is determined that the system is in a heating mode.