The overgrowth of K-beta"-Al2O3 on single-crystal sapphire substrates has been investigated using carbonate, chloride and beta"(beta)-Al2O3 vapour sources at temperatures between 950 degrees C and 1250 degrees C. Single-crystal film is obtained on the sapphire substrate using a (K, Li)-beta"-Al2O3 powder vapour source at 1120-1250 degrees C. The mechanism of growth and the chemical and crystallographic factors involved are discussed.