We present the final data from the spectroscopic survey of the ROSAT-ESO Flux-Limited X-ray (REFLEX) catalog of galaxy clusters. The REFLEX survey covers 4.24 steradians (34% of the entire sky) below a declination of 2.5 deg and at high Galactic latitude (|b| > 20 deg). The REFLEX catalog includes 447 entries with a median redshift of 0.08 and is better than 90% complete to a limiting flux fx = 3x10^{-12} erg s^{-1} cm^{-2} (0.1 to 2.4 keV), representing the largest statistically homogeneous sample of clusters drawn from the ROSAT All-Sky Survey (RASS) to date. Here we describe the details of the spectroscopic observations carried out at the ESO 1.5 m, 2.2 m, and 3.6 m telescopes, as well as the data reduction and redshift measurement techniques. The spectra typically cover the wavelength range 3600-7500 A at a FWHM resolution of ~14 A, and the measured redshifts have a total rms error of ~100 km s^{-1}. In total we present 1406 new galaxy redshifts in 192 clusters, most of which previously did not have any redshift measured. Finally, the luminosity/redshift distributions of the cluster sample and a comparison to the no-evolution expectations from the cluster X-ray luminosity function are presented.
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