  We describe the selection of an X-ray flux-limited sample of bright clusters
of galaxies in the southern hemisphere, based on the first analysis of the
ROSAT All-Sky Survey data (RASS1). The sample is constructed starting from an
identification of candidate clusters in RASS1, and their X-ray fluxes are
remeasured using the Steepness Ratio Technique. This method is better suited
than the RASS1 standard algorithm for measuring flux from extended sources. The
final sample is count-rate-limited in the ROSAT hard band (0.5-2.0 keV), so
that due to the distribution of NH, its effective flux limit varies between
about 3-4 x 10**-12 ergs cm**-2 s**-1 over the selected area. This covers the
Decl<2.5 deg part of the south Galactic cap region (b<-20 deg) - with the
exclusion of patches of low RASS1 exposure time and of the Magellanic Clouds
area - for a total of 8235 deg**2. 130 candidate sources fulfill our selection
criteria for bonafide clusters of galaxies in this area. Of these, 101 are
Abell/ACO clusters, while 29 do not have a counterpart in these catalogs. Of
these clusters, 126 (97%) clusters have a redshift and for these we compute an
X-ray luminosity. 20% of the cluster redshifts come from new observations, as
part of the ESO Key Program REFLEX Cluster Survey that is under completion.
Considering the intrinsic biases and incompletenesses introduced by the flux
selection and source identification processes, we estimate the overall
completeness to be better than 90%. The observed number count distribution,
LogN-LogS, is well fitted by a power law with slope alpha = 1.34 +/- 0.15 and
normalization A = 11.87 +/- 1.04 sr**-1 (10**-11 ergs cm**-2 s**-1)**alpha, in
good agreement with other measurements.
