We report the growth of high quality AlGaN/GaN heterostructures on sapphire by ammonia gas-source molecular-beam epitaxy. By inserting a low-temperature-grown AIN interlayer during the growth Of the thick GaN buffer, the piezoelectrically-induced two-dimensional electron gas (2DEG) in the unintentional doped AlGaN/GaN heterostructurcs exhibited substantially increased mobility. With the optimized AlN interlayer thickness of 30 nm, electron mobilities as high as 1500, 10310, and 12000 cm(2)/Vs were obtained at room temperature, 77 and 0.3 K respectively. Sheet densities of 9 X 10(12) and 6 x 10(12) cm(-2) were measured at room temperature and 77 K, respectively. The 2DEG was confirmed by strong and well resolved Shubnikov-de Haas oscillations starting at 2.8 T. Photoluminescence measurements and atomic force microscopy revealed that the densities of native donors and grain boundaries were effectively reduced in the AlGaN/GaN heterostructures with AlN interlayers.