High quality AlGaN/GaN heterostructures have been grown by rf plasma-assisted molecular beam epitaxy (MBE) on n-type GaN templates grown on sapphire by metalorganic chemical vapor deposition (MOCVD). By optimizing the AlGaN thickness and the Al content, record low temperature mobilities have been achieved. Temperature and magnetic field dependent Hall effect, and Shubnikov-de Haas oscillations, were used to probe the two-dimensional electron gas (2DEG) with consistent results. The unintentionally doped Al0.09Ga0.91N heterostructures exhibit a measured 77 K Hall mobility of 24,000 cm2/V·s (nsh=2.5×1012), 12 K mobility of 52,000 cm2/V·s and ∼4 K mobility of 60,000 cm2/V·s (nsh=2.25×1012), all records for this material system. The magnetic field dependent Hall effect revealed carriers from the bulk GaN freeze out and that a single carrier system is dominant below 80 K.