NOVELTY - The apparatus (101) has an off-gas line (L3) into which the off-gas of a methane separation apparatus (1) flows in, and a burner (4) which burns off the off-gas. A bypass line (L1b) bypasses the methane separation apparatus and supplies biogas to the burner. A controller (V1) controls the supply amount of biogas which flows through the bypass line, and the detection apparatuses (31,41) detect the combustion state of the burner. The supply amount of biogas is reduced when the combustion state of the burner is stable or else the supply amount of biogas is increased.    USE - Off-gas combustion apparatus used for separating methane (CH4) from biogas, using pressure swing adsorption (PSA) method. Can also be used for separating methane from biogas, using membrane separation method.    ADVANTAGE - The diffusion of separated methane into the atmosphere can be prevented completely, and stabilized combustion is made possible without the danger of misfiring, even if sufficient amount of supplementary fuels such as town gas is not supplied to the burner. The off-gas can be burnt reliably and the environment is effectively prevented from being contaminated. The combustion catalyst can always be maintained at high temperature, except at the time of starting, and low concentration methane can be burnt reliably according to the catalyst.    DETAILED DESCRIPTION - An off-gas line branching from the specified off-gas line is connected to a combustion catalyst and to the burner. A switching apparatus is provided at the junction of the two off-gas lines, and a heating apparatus which heats the off-gas supplied to the catalyst is arranged between a heat exchanger (5) and the catalyst. A temperature measuring device is set to measure the temperature of off-gas supplied to the catalyst and the surface temperature of catalyst. A waste gas line (Lx) is formed for the flow of waste gas of catalyst to a combustion exhaust gas line. An air supply apparatus is set to supply combustion air to the catalyst, through an air supply line (La). A combustion exhaust gas line is connected to the burner.    DESCRIPTION OF DRAWING(S) - The drawing shows a schematic block diagram of the off-gas combustion apparatus. (Drawing includes non-English language text)    Methane separation apparatus (1)    Burner (4)    Heat exchanger (5)    Detection apparatuses (31,41)    Off-gas combustion apparatus (101)    Bypass line (L1b)    Off-gas line (L3)    Air supply line (La)    Waste gas line (Lx)    Controller (V1)