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 re-measured 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 N _ { H } , its effective flux limit varies between \sim 3. and 4. \times 10 ^ { -12 } erg cm ^ { -2 } s ^ { -1 } over the selected area . This covers the \delta < 2.5 ^ { o } part of the south Galactic cap region ( b _ { II } < -20 ^ { o } ) – 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 . 101 of these are Abell/ACO clusters , while 29 do not have a counterpart in these catalogs . 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 \pm 0.15 and normalization A = 11.87 \pm 1.04 sr ^ { -1 } ( 10 ^ { -11 } erg cm ^ { -2 } s ^ { -1 } ) ^ { \alpha } , in good agreement with other measurements .