The structure of binary aluminosilicate glasses containing 60-67 mol % Al2O3 were investigated using high-resolution Al-27 NMR and X-ray and neutron diffraction. The glasses were made by aerodynamic levitation of molten oxides. The 67% alumina composition required a cooling rate of similar to 1600 degrees C s(1-) to form glass from submillimeter sized samples. NMR results show that the glasses contain aluminum in 4-, 5-, and 6-fold coordination in the approximate ratio 4:5:1. The average Al coordination increases from 4.57 to 4.73 as the fraction of octahedral Al increases with alumina content. The diffraction results on the 67% composition are consistent with a disordered Al framework with Al ions in a range of coordination environments that are substantially different from those found in the equilibrium crystalline phases. Analysis of the neutron and X-ray structure factors yields an average bond angle of 125 +/- 4 degrees between an Al ion and the adjoining cation via a bridging oxygen. We propose that the structure of the glass is a "transition state" between the alumina-rich liquid and the equilibrium mullite phase that are dominated by 4- and 6-coordinated aluminum ions, respectively.