In order to overcome the limitations and problems that earlier methods have experienced, a method for operating a burner used in the thermal step of a Claus process is disclosed. The burner has at least one combustion air port, at least one oxygen port, and at least one fuel port. The oxidant flow is made up of an oxygen flow of technical pure oxygen with an oxygen concentration of at least 90 vol. %, preferably at least 99 vol. %, and if necessary of a second gas flow, and it is determined whether the oxygen flow is below a pre-set minimum flow and if the oxygen flow is below a pre-set minimum flow the air flow is split into a main air flow and a side air flow and the oxygen flow is combined with the side air flow creating an oxygen-enriched side air flow which is fed to the oxygen port. A corresponding burner is also disclosed.