In a high efficiency regenerator burner for heating spaces, the exhaust gas generated by the burner is provided which is conducted alternately through different regenerator cartridges and a partial stream of the exhaust gas is conducted under the control of an orifice plate through a bypass space in which the regenerator cartridges are disposed. A control structure is disposed in a burner head for controlling the exhaust gas bypass flow volume and also to control the main exhaust gas flow as well as the combustion air flow through the regenerator cartridges.