We describe the fringe-packet tracking system used to equalize the optical path lengths at the Infrared Optical Telescope Array interferometer. The measurement of closure phases requires obtaining fringes on three baselines simultaneously. This is accomplished by use of an algorithm based on double Fourier interferometry for obtaining the wavelength-dependent phase of the fringes and a group-delay tracking algorithm for determining the position of the fringe packet. A comparison of data acquired with and without the fringe-packet tracker shows a factor of ~3 reduction of the error in the closure-phase measurement. The fringe-packet tracker has been able so far to track fringes with signal-to-noise ratios as low as 1.8 for stars as faint as m_H=7.0.