A burner for operating a heat generator, includes a swirl generator (100) for a combustion air flow (115) means for injecting at least one fuel into the combustion air flow. A mixing section is arranged downstream of this swirl generator, which mixing section has, within a first section in the flow direction, a number of ducts in a transition piece (200) for the transfer of a flow formed in the swirl generator into a mixing tube (20) connected downstream of this transition piece. A combustion space (30) is arranged downstream of this mixing tube. The swirl generator (100) is integrated into an inlet flow cross section (10) which is essentially closed by the swirl generator, and is subjected to the combustion air flow (115).