An SOx trap catalyst able to trap SOx contained in the exhaust gas is arranged in an engine exhaust passage upstream of an NOx storing catalyst in an internal combustion engine. When the SOx trap rate of the SOx trap catalyst falls, fuel is added in the exhaust gas flowing into the SOx trap catalyst to form in the SOx trap catalyst a region in which an air-fuel ratio becomes locally rich. SOx released from the SOx trap catalyst in this region can be trapped in the SOx trap catalyst at the downstream side once again without flowing out from the downstream end of the SOx trap catalyst.