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We report the use of suboxide molecular-beam epitaxy (S-MBE) to grow α-(AlxGa1−x)2O3 films on (110) sapphire substrates over the 0 < x < 0.95 range of aluminum content. In S-MBE, 99.98% of the gallium-containing molecular beam arrives at the substrate in a preoxidized form as gallium suboxide (Ga2O). This bypasses the rate-limiting step of conventional MBE for the growth of gallium oxide (Ga2O3) from a gallium molecular beam and allows us to grow fully epitaxial α-(AlxGa1−x)2O3 films at growth rates exceeding 1 µm/h and relatively low substrate temperature (Tsub = 605 ± 15 °C). The ability to grow α-(AlxGa1−x)2O3 over the nominally full composition range is confirmed by Vegard’s law applied to the x-ray diffraction data and by optical bandgap measurements with ultraviolet–visible spectroscopy. We show that S-MBE allows straightforward composition control and bandgap selection for α-(AlxGa1−x)2O3 films as the aluminum incorporation x in the film is linear with the relative flux ratio of aluminum to Ga2O. The films are characterized by atomic-force microscopy, x-ray diffraction, and scanning transmission electron microscopy (STEM). These α-(AlxGa1−x)2O3 films grown by S-MBE at record growth rates exhibit a rocking curve full width at half maximum of ≊ 12 arc secs, rms roughness <1 nm, and are fully commensurate for x ≥ 0.5 for 20–50 nm thick films. STEM imaging of the x = 0.78 sample reveals high structural quality and uniform composition. Despite the high structural quality of the films, our attempts at doping with silicon result in highly insulating films.