The Monitored Drift Tube (MDT) chambers for the muon spectrometer of the ATLAS detector at the Large Hadron Collider (LHC) consist of 3-4 layers of pressurised drift tubes on either side of a space frame carrying an optical deformation monitoring system. The chambers have to provide a track position resolution of 40 microns with a single-tube resolution of at least 80 microns and a sense wire positioning accu- racy of 20 ?microns (rms). The feasibility was demonstrated with the full-scale prototype of one of the largest MDT chambers with 432 drift tubes of 3.8 m length. For the ATLAS muon spectrometer, 88 chambers of this type have to be built. The first chamber has been completed with a wire positioning accuracy of 14 microns (rms).