An internal combustion engine air-fuel ratio control apparatus, comprising: an exhaust-gas purification catalyst unit that is provided in an exhaust passage of an internal combustion engine and has an oxygen storage capability storing oxygen in exhaust gas when the oxygen concentration in exhaust gas flowing into the exhaust-gas purification catalyst unit is excessively high and releasing oxygen stored in the exhaust gas-purification catalyst unit when the oxygen concentration in the exhaust gas is excessively low; an intake-air amount detector that detects an amount of intake air drawn into the internal combustion engine; a linear air-fuel ratio sensor that is provided upstream of the exhaust-gas purification catalyst unit and has an output characteristic that is substantially proportional to the air-fuel ratio of exhaust gas; an oxygen sensor that is provided downstream of the exhaust-gas purification catalyst unit and detects whether the air-fuel ratio of exhaust gas is rich or lean; a target air-fuel ratio controller that executes feedback control of a target air-fuel ratio of exhaust gas flowing into the exhaust-gas purification catalyst unit based on an outputof the intake-air amount detector and an output of the oxygen sensor: and a fuel-injection amount controller that executes feedback control of a fuel injection amount based on an output of the linear air-fuel ratio sensor such that the air-fuel ratio of the exhaust gas flowing into the exhaust-gas purification catalyst unit equals to the target air-fuel ratio, wherein the target air-fuel ratio controller implements a first target air-fuel ratio feedback control mode in which the target air-fuel ratio is controlled through at least proportional-integral control such that, even if the intake air amount changes, the amount by which the amount of oxygen in the exhaust-gas purification catalyst unit is corrected per unit time is maintained at a constant level, a proportional correction term for the proportional-integral control of the first target air-fuel ratio feedback control mode is multiplied by a first correction coefficient that is made smaller the larger the intake air amount; an integral correction term for the proportional-integral control of the first target air-fuel ratio feedback control mode is multiplied by a second correction coefficient that is made larger the larger the intake air amount, and the target air-fuel ratio controller controls the target air-fuel ratio so as to suppress an increase in the air-fuel ratio of atmosphere in the exhaust-gas purification catalyst unit when the intake air amount is smaller than a predetermined amount and the air-fuel ratio detected by the oxygen sensor is rich. an exhaust-gas purification catalyst unit that is provided in an exhaust passage of an internal combustion engine and has an oxygen storage capability storing oxygen in exhaust gas when the oxygen concentration in exhaust gas flowing into the exhaust-gas purification catalyst unit is excessively high and releasing oxygen stored in the exhaust gas-purification catalyst unit when the oxygen concentration in the exhaust gas is excessively low; an intake-air amount detector that detects an amount of intake air drawn into the internal combustion engine; a linear air-fuel ratio sensor that is provided upstream of the exhaust-gas purification catalyst unit and has an output characteristic that is substantially proportional to the air-fuel ratio of exhaust gas; an oxygen sensor that is provided downstream of the exhaust-gas purification catalyst unit and detects whether the air-fuel ratio of exhaust gas is rich or lean; a target air-fuel ratio controller that executes feedback control of a target air-fuel ratio of exhaust gas flowing into the exhaust-gas purification catalyst unit based on an outputof the intake-air amount detector and an output of the oxygen sensor: and a fuel-injection amount controller that executes feedback control of a fuel injection amount based on an output of the linear air-fuel ratio sensor such that the air-fuel ratio of the exhaust gas flowing into the exhaust-gas purification catalyst unit equals to the target air-fuel ratio, wherein