The invention relates to a process for the gasification of a carbonaceous fuel in a fluidized bed gasifier (1). In the process, a fuel (5), such as coal or peat, and a primary gasification gas (6, 7), such as air, are fed into the gasifier, and a product gas (10) is removed from the gasifier, the product gas being purified by separating fine-grained solid ingredients (16) therefrom. The solids fraction (16) which is separated from the product gas (16) and which contains unreacted carbon is fed together with a stoichiometric excess of air (27) into another fluidized bed reactor (20), which serves as an oxidizer, and the oxygen-containing gas (34) exiting from the oxidizer is directed to the fluidized bed gasifier (1) as a secondary gasification gas. By the increased oxidation, the carbon conversion in the process is increased and a residual ash with a low carbon content is produced.