Previously, in order to produce a dimensionally stable fused-quartz hollow cylinder, an output cylinder having a centre axis would have been prepared, said output cylinder being produced by means of a bore having a boring bar, said bore rotating about a horizontal axis of rotation, and by means of a boring head that has an internal bore and is fixed to said bore in a torsionally rigid manner, wherein the boring head adopts a continually changing boring head position that is continuously ascertained by means of a measuring device and is returned to a target position in the event of any deviation. The design outlay is high, however. The problem addressed by the invention is therefore that of providing a cost-effective method for producing a dimensionally stable fused-quartz hollow cylinder.