The piezoelectric potential accompanying a surface acoustic wave ͑SAW͒ launched on the surface of a GaAs/Al x Ga 1Ϫx As heterostructure is calculated. The screening due to a narrow metal gate on the surface is included and a closed form analytic solution for the SAW potential is obtained. This potential is used to calculate the SAW-induced acoustoelectric current in a quasi-one-dimensional electron channel formed in the heterostructure by the split gates. At gate voltages beyond the pinch off, electrons are transported through the channel in local quantum wells formed by the SAW potential. In recent experiments, Talyanskii et al. ͓Phys. Rev. B 56, 15 180 ͑1997͔͒ found that, in this regime, the acoustoelectric current I versus the gate voltage displays a step-like behavior. The values of the current on the plateaus are quantized with Iϭne f where e is electron charge, f is the SAW frequency, and n is the number of electrons transported through the channel per SAW cycle. Using a simple model for the electrostatic barrier in the channel, we show that when one electron is trapped in the quantum well, the current depends on a parameter  defined as the ratio of the piezoelectric potential amplitude to the height of the electrostatic barrier. When  increases, the current rapidly increases from zero to its quantized value Iϭe f . The obtained results explain the first current plateau and qualitatively agree very well with the experiment. ͓S0163-1829͑98͒00540-2͔