The invention relates to a burner, in particular a residual gas burner for a fuel cell system. The burner includes a combustion chamber which is bordered by a supply wall and by a heat exchanger and which is encompassed at the sides by a burner wall. The heat exchanger is a cross-current heat exchanger having a primary path and a secondary path. The supply wall has a burner zone with oxidizer openings for oxidizer gas and with combustion gas openings for combustion gas and a bypass zone with bypass openings for bypass gas. The bypass zone is arranged in a section of the supply wall which is allocated to an area of the heat exchanger adjacent to the primary path and to the secondary path at the inlet end, so that the bypass gas or a bypass gas-burner exhaust gas mixture acts upon this area.