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An essential part of the Muon Spectrometer of the ATLAS experiment is based on the Monitored Drift Tube (MDT) technology. About 1200 muon drift chambers are being built at 13 institutes all over the world. The MDT chambers require an exceptional mechanical construction accuracy of better than 20 μm. A dedicated X-ray tomograph has been developed at CERN since 1996 to control the mechanical quality of the chambers. The chamber wire positions are measured with a statistical error of 2 μm and a systematic error of 2 μm over a working area of 2.2 m × 0.6 m. During the construction phase of the MDT chambers, from middle 2000 to mid 2005, the X-ray tomograph is the key tool for ensuring consistent chamber production with a sampling rate of ∼15%. To achieve this program efficiently, an effort for complete automation of the tomograph operation has been underway. Until September 2003, 79 chambers out of 739 chambers produced at 11 of the construction sites have been measured. The X-ray tomograph has proved to be an essential and powerful tool in assessing the validity of the various construction steps.