To make a cutting quality satisfactory at the time of cutting a thick plate and to make it possible to perform an effective cutting operation at an inexpensive running cost.

In the thermal cutting machine using the oxygen gas of an oxygen cylinder on the market as the assist gas, a pressure fluctuation adsorption system oxygen gas refiner 23 refining the oxygen gas of the oxygen cylinder for the assist gas to the high purity of a prescribed value of over and to an oxygen concentration with specific purity, is provided. Also, a reservoir tank 24 storing temporarily the refined oxygen gas and a regulator for the pressure adjustment of the assist gas to be supplied to a nozzle 7, may be provided. Also, when the oxygen concentration of the output gas of the oxygen cylinder is a prescribed value or over, this oxygen gas is directly used, and when the oxygen concentration is lass than the prescribed value, this oxygen gas is used as the assist gas after refining it by the oxygen gas refiner 23.