In a method of operating a low-pollution premixing burner (2) stabilized by means of vortex breakdown, in particular a burner of the double-cone type of construction, with gaseous fuels (4, 10), the main fuel gas (4) being fed to the burner (2) via a main gas tube (3) connected in one piece to the burner (2) and the pilot gas (10) being fed to the burner (2) near the axis of the latter via a separate feed line (9) by means of an exchangeably inserted fuel lance (8), and the pilot gas (10) being mixed inside the fuel lance (8) with air (17) fed from a plenum (16) outside the burner hood (6), the pilot-gas/air mixture (25) is fed to a catalyzer (21) arranged inside the fuel lance (8) at the tip of the burner (2) and is ignited and burnt there. The hot gas flow is then mixed with the colder main burner flow in the burner interior space (14).