A device and a method for flame stabilization in a burner (10), includes a burner housing at least partially enclosing a burner volume, into which may be introduced via at least one fuel line, fuel, and via at least one air feed means, air, forming an air / fuel mixture spreading in a preferred flow direction, which may be ignited in a combustion chamber (11) connecting downstream of the burner housing to form a stationary flame (13). Upstream of the flame (13), a catalyst arrangement (1) is provided through which an air / pilot fuel mixture (4), separate from the air / fuel mixture, is flowable. The catalyst arrangement (1) has at least two catalyst stages which are located one behind the other in the through-flow direction, of which the catalyst stage (3) located upstream, the so-called POX-catalyst, is flow-washable by the air / pilot fuel mixture (4) with an air / pilot fuel mixture ratio λ<1, by which catalyst stage (3) the air / pilot fuel mixture (4) is partially oxidized, and of which catalyst stages the downstream catalyst stage (8), the so-called FOX-catalyst, is flow-washable by a leaned air / pilot fuel mixture (7) with a mixture ratio λ>1, by which the leaned air / pilot fuel mixture is completely oxidized forming an inert hot gas flow (9).