In an internal combustion engine, an SOX trap catalyst (13) which traps SOX contained in the exhaust gas is arranged upstream of an NOX storing catalyst (15) in an engine exhaust passage. When the NOX storing catalyst (15) should release NOX, auxiliary fuel is injected into a combustion chamber (2) at a combustible timing after the completion of injection of fuel into the combustion chamber (2) for generating the engine output. At this time, the auxiliary fuel is injected so that the air-fuel ratio A/F of the combustion gas in the combustion chamber (2) does not become below a minimum allowable air-fuel ratio M where SOX trapped at the SOX trap catalyst (13) will not be released.