We report the use of suboxide molecular-beam epitaxy (S-MBE) to grow β-Ga2O3 at a growth rate of ~1 µm/h with control of the silicon doping concentration from 5x10^16 to 10^19 cm^-3. In S-MBE, pre-oxidized gallium in the form of a molecular beam that is 99.98% Ga2O, i.e., gallium suboxide, is supplied. Directly supplying Ga2O to the growth surface bypasses the rate-limiting first step of the two-step reaction mechanism involved in the growth of β-Ga2O3 by conventional MBE. As a result, a growth rate of ~1 µm/h is readily achieved at a relatively low growth temperature (Tsub ≈ 525 °C), resulting in films with high structural perfection and smooth surfaces (rms roughness of < 2 nm on ~1 µm thick films). Silicon-containing oxide sources (SiO and SiO2) producing an SiO suboxide molecular beam are used to dope the β Ga2O3 layers. Temperature-dependent Hall effect measurements on a 1 µm thick film with a mobile carrier concentration of 2.7x10^17 cm^–3 reveal a room-temperature mobility of 124 cm^2 V^–1 s^–1 that increases to 627 cm^2 V^–1 s^–1 at 76 K; the silicon dopants are found to exhibit an activation energy of 27 meV. We also demonstrate working MESFETs made from these silicon-doped β-Ga2O3 films grown by S-MBE at growth rates of ~1 μm/h.