When the temperature of the exhaust gas flowing into an NSR catalyst exceeds a specific threshold temperature that is determined on the basis of the cetane number of fuel in such a way that the specific threshold temperature is set lower when the cetane number of the fuel is low than when it is high, fuel is supplied to the exhaust gas by fuel supply device to perform an NOX reduction process for the NSR catalyst. If the quantity of heat generated in the NSR catalyst per unit time is smaller than a specific value while the NOX reduction process is being performed, the NOX reduction process in progress is suspended, and the NOX reduction process is performed later on when the temperature of the exhaust gas flowing into the NSR catalyst exceeds an updated threshold temperature higher than the specific threshold temperature.