We present a catalogue of accurate positions and correlated flux densities for 410 flat-spectrum, compact extragalactic radio sources previously detected in the AT20G survey. The catalogue spans the declination range [-90{deg}, -40{deg}] and was constructed from four 24-hour VLBI observing sessions with the Australian Long Baseline Array at 8.3GHz. The VLBI detection rate in these experiments is 97%, the median uncertainty of the source positions is 2.6mas, and the median correlated flux density on projected baselines longer than 1000km is 0.14Jy. The goals of this work are 1) to provide a pool of southern sources with positions accurate to a few milliarcsec, which can be used for phase referencing observations, geodetic VLBI and space navigation; 2) to extend the complete flux-limited sample of compact extragalactic sources to the southern hemisphere; and 3) to investigate the parsec-scale properties of high-frequency selected sources from the AT20G survey. As a result of this VLBI campaign, the number of compact radio sources south of declination -40 deg which have measured VLBI correlated flux densities and positions known to milliarcsec accuracy has increased by a factor of 3.5. The catalogue and supporting material is available at http://astrogeo.org/lcs1.