An apparatus for controlling an air-fuel ratio of an internal-combustion engine includes an air-fuel ratio detector, a fluctuation signal generating device, an air-fuel ratio fluctuation device, a 0.5th-order frequency component strength calculator, a fluctuation frequency component strength calculator, a reference component strength calculator, and an imbalance fault determining device. The reference component strength calculator is configured to calculate strength of a reference component in accordance with strength of a first frequency component and strength of a second frequency component. The imbalance fault determining device is configured to make a determination of an imbalance fault in which air-fuel ratios of a plurality of cylinders vary beyond a tolerance limit on a basis of a relative relationship between strength of the 0.5th-order frequency component and the strength of the reference component.