A premix burner has a mixing section (3) for a heat generator, sectional conical shells (5) which complement one another to form a swirl body, enclose a conically widening swirl space (6), and mutually define tangential air-inlet slots (7), along which feeds (8) for gaseous fuel are provided in a distributed manner, having at least one fuel feed (11) for liquid fuel, this fuel feed (11) being arranged along a burner axis (A) passing centrally through the swirl space (6), and having a mixing tube (4) adjoining the swirl body downstream via a transition piece (2). At least one additional fuel feed (13) for liquid fuel is provided in the region of the swirl body, the transition piece (2), and / or the mixing tube (4).