This paper presents the third extension to the Very Large Baseline Array (VLBA) Calibrator Survey, containing 360 new sources not previously observed with very long baseline interferometry (VLBI). The survey, based on three 24 hour VLBA observing sessions, fills the areas on the sky above declination -45 degr where the calibrator density is less than one source within a 4 degr radius disk at any given direction. The positions were derived from astrometric analysis of the group delays determined at 2.3 and 8.4 GHz frequency bands using the Calc/Solve software package. The VCS3 catalogue of source positions, plots of correlated flux density versus the length of projected baseline, contour plots and fits files of naturally weighted CLEAN images as well as calibrated visibility function files are available on the Web at http://gemini.gsfc.nasa.gov/vcs3