A cosmic ray measurement facility has been set up at Munich University and is used at present to commission and calibrate monitored drift tube (MDT) chambers for the muon spectrometer of the ATLAS experiment. Each tested chamber - produced in collaboration with the Max-Planck-Institut fur Physik and the Joint Institute for Nuclear Research in Dubna - consists of 432 drift tubes with a diameter of 3 cm, which are arranged in 2 /spl times/ 3 layers, and measure 3.8 m /spl times/ 2.2 m /spl times/ 0.5 m. The response of all drift tubes and its homogeneity across the chamber is measured in the cosmic ray facility. Two MDT chambers which were precisely mapped with an X-ray tomograph provide reference tracking for the enclosed third chamber which is to be calibrated. The sense wire positions are determined from a comparison of its drift time measurements with the reference tracks. The geometry of the tested chamber - the grid constants of the drift tubes in a layer, the distances and tilt angles of the layers - can then be derived from these positions. Alignment systems are used to monitor chamber movements at the micrometer level. In the ongoing series commissioning, a rate of 1 chamber per week has been reached. The parameters describing the geometry of the MDT chambers are determined with a precision in the 10 /spl mu/m range. The measurements with cosmic muons complement the chamber surveys performed during construction and will provide an important input for the calibration of the entire ATLAS muon spectrometer.