Construction and Test of a Full Prototype Drift-Tube Chamber for the Upgrade of the ATLAS Muon Spectrometer at High LHC Luminosities B.Bittner, J.Dubbert, O.Kortner, H.Kroha, A.Manfredini, S.Ott, R.Richter, Ph.Schwegler, D.Zanzi, Max-Planck-Institut fuer Physik, Munich O.Biebel, A.Engl, R.Hertenberger, A.Zibell Ludwig-Maximilians University, Munich For the planned high-luminosity upgrades of the Large Hadron Collider (LHC) background rates of neutrons and gamma rays of up to 14 kHz/cm^2 are expected which exceed the rate capability of the current ATLAS precision muon tracking detectors, the Monitored Drift Tube (MDT) chambers, with a drift tube diameter of 30 mm. So called sMDT chambers with a drift tube diameter of 15 mm have been developed for upgrades of the ATLAS muon spectrometer. A full sMDT prototype chamber has been constructed and tested in a muon beam at CERN and with cosmic muons at high gamma irradiation rates of up to 23 kHz/cm^2. The test results demonstrate the required track reconstruction efficiency and spatial resolution of the sMDT chambers at background rates well beyond the maximum expected value. The sense wire positions in the prototype chamber have been measured with about 5 micron precision with cosmic rays Tests at the CERN Gamma Irradiation Facility showed that drift-tube detectors with 15 mm diameter aluminum tubes operated with Ar:CO2 (93:7) gas at 3 bar and a maximum drift time of about 200 ns provide efficient and high-resolution muon tracking up to the highest expected rates. For 15 mm tube diameter, space charge effects deteriorating the spatial resolution of larger diameter tubes are strongly suppressed. The sense wires have to be positioned in the chamber with an accuracy of better than 50 micron in order to achieve a chamber resolution of about 50 micron. We report about the design, construction and test of a prototype chamber which fulfills these requirements.