In a known method for the manufacture of a tube made of a vitreous material, especially of quartz glass, a hollow cylindrical semifinished product made of a vitreous material is carried essentially vertically to a heating zone, wherein it is heated and drawn off downwards—without the use of tools—to the tube by forming a transitional area from semifinished product to tube, while diameter and wall thickness of the tube are continuously measured, and the tube's measured geometrical data being used to generate a control signal with the aid of which a pressure difference is regulated between pressure P1 in the interior space of the semifinished product, the transitional area and the tube, as well as pressure P2 in the heating chamber which is regulated in the heating zone at least in the transitional area from semifinished product to tube and its adjacent tube area. This method is to be developed further so as to enable the processing of a broader spectrum of hollow cylindrical semifinished products to tubes with the desired inside and outside dimensions. This task is solved according to the invention such that for the manufacture of tubes with the ratio of their outside diameter (DRa) to their inside diameter (DRi) being greater than the ratio of the outside diameter (DHa) to the inside diameter (DHi) of the semifinished product—pressure P1 in the inside space of the semifinished product, the transitional area and the tube being kept at a value which is smaller than the value of pressure P2 in the heating chamber, with pressure P1 being maintained by means of a suction (vacuum) pump.