In the process a glass tube line is drawn from a glass tube production plant and the glass tube is severed from the glass tube line. At the same time the still heated glass tube is subjected to a selective local heat treatment of the at least one tube end section prior to cooling down completely in order to reduce the stresses in the respective tube end section. Not the entire glass tube but instead only the respective tube end section is subjected to the selective local heat treatment. The heat treatment is performed in particular in such a manner that the mechanical tensile stress in the region between 7 mm and 20 mm from the tube end or ends is reduced to below 6.0 MPa, more preferably to below 4.5 MPa.