A premix burner consisting essentially of a swirl generator for a combustion air stream or another gaseous oxidizing medium and of conduits for injecting fuel into the swirl flow thus generated. As a result of a discontinuous cross-sectional widening at the transition to the combustion space, the swirl flow bursts open and a backflow bubble is formed, which serves for flame stabilization. For use under extreme thermal conditions, the burner front is provided with a heat shield which is fastened to a carrier structure in such a way as to allow, as unimpeded as possible, differential thermal expansions. The carrier structure is preferably provided with orifices, through which cooling air for the impact cooling of the heat shield flows. In a preferred variant, the fuel gas supply is designed as pipes which are led through orifices of the swirl generator without any fixed connection. As a result of this design, thermal stresses in the material of the burner are avoided as far as possible.