A method for producing a synthesis-gas product gas and a vapor stream includes catalytic steam reforming a hydrocarbonaceous feedstock in a steam reformer. The hot synthesis-gas product gas stream is cooled in a heat exchanger to form a cooled synthesis-gas product gas stream and a first partial vapor stream, which is supplied to the product vapor stream. The reforming furnace is operated so as to burn a burner feedstock in burners, cool a hot flue gas stream from the burners in a heat exchanger to form a cooled flue gas stream and a second partial vapor stream, and separate the cooled flue gas stream into a waste gas stream and a flue gas recirculation stream. The flow of the recirculated flue gas is increased with decreasing flow of the synthesis-gas product gas to obtain an approximately constant product vapor stream by increasing the second partial vapor stream.