We present the catalogue of the REFLEX Cluster Survey providing information on the X-ray properties, redshifts, and some identification details of the clusters in the REFLEX sample. The catalogue describes a statistically complete X-ray flux-limited sample of 447 galaxy clusters above an X-ray flux of 3 x 10 super(-12) erg s super(-1) cm super(-2) (0.1 to 2.4 keV) in an area of 4.24 ster in the southern sky. The cluster candidates were first selected by their X-ray emission in the ROSAT-All Sky Survey and subsequently spectroscopically identified in the frame of an ESO key programme. Previously described tests have shown that the sample is more than 90% complete and there is a conservative upper limit of 9% on the fraction of clusters with a dominant X-ray contamination from AGN. In addition to the cluster catalogue we also describe the complete selection criteria as a function of the sky position and the conversion functions used to analyse the X-ray data. These are essential for the precise statistical analysis of the large-scale cluster distribution. This data set is at present the largest, statistically complete X-ray galaxy cluster sample. Together with these data set we also provide for the first time the full three-dimensional selection function. The sample forms the basis of several cosmological studies, one of the most important applications being the assessment of the statistics of the large-scale structure of the universe and the test of cosmological models. Part of these cosmological results have already been published.