A method for diagnosing the degree of deterioration of a catalyst disposed in an exhaust path of an internal combustion engine and oxidizing or adsorbing a target gas, including at least one of a hydrocarbon gas and a carbon monoxide gas, in an exhaust gas from the internal combustion engine, is adapted to determine whether deterioration exceeding an acceptable level of a catalyst occurs or not by comparing, at any timing when the internal combustion engine is in a state of a steady operation, the concentration of a target gas detected downstream from the catalyst in the exhaust path with a threshold value of the concentration of said target gas corresponding to the temperature of said catalyst at the timing which is previously defined according to an allowable range of an index value representing the degree of oxidation or adsorption at the catalyst corresponding to the temperature of said catalyst at the timing.