Exhaust emission control apparatus of internal combustion engine and control method of the same

The exhaust emission control apparatus is provided with a NOx catalyst (23) for purifying exhaust gas flowing out of an internal combustion engine (1) and a NOx sensor (29) for detecting a NOx concentration in exhaust gas flowing out of the NOx catalyst. The internal combustion engine (1) and the exhaust emission control apparatus are mounted on a vehicle. The exhaust emission control apparatus calculates a total outflow amount of NOx flowing out of the NOx catalyst (23) obtained while the vehicle runs for a predetermined distance based on a NOx concentration detected by the above-mentioned sensor (29), calculates a NOx outflow amount per unit running distance based on the total NOx outflow amount, and supplies the NOx catalyst (23) with exhaust gas with a rich air-fuel ratio when a NOx outflow amount per unit running distance exceeds a predetermined amount.