The extended ROSAT-ESO Flux Limited X-ray Galaxy Cluster Survey (REFLEX
  II)\\ II. Construction and Properties of the Survey
  Galaxy clusters provide unique laboratories to study astrophysical processes
on large scales and are important probes for cosmology. X-ray observations are
currently the best means of detecting and characterizing galaxy clusters. In
this paper we describe the construction of the REFLEX II galaxy cluster survey
based on the southern part of the ROSAT All-Sky Survey. REFLEX II extends the
REFLEX I survey by a factor of about two down to a flux limit of $1.8 \times
10^{-12}$ erg s$^{-1}$ cm$^{-2}$ (0.1 - 2.4 keV). We describe the determination
of the X-ray parameters, the process of X-ray source identification, and the
construction of the survey selection function. The REFLEX II cluster sample
comprises currently 915 objects. A standard selection function is derived for a
lower source count limit of 20 photons in addition to the flux limit. The
median redshift of the sample is $z = 0.102$. Internal consistency checks and
the comparison to several other galaxy cluster surveys imply that REFLEX II is
better than 90\% complete with a contamination less than 10\%.
