The multi-frequency Sedentary Survey is a flux limited, statistically well-defined sample of highly X-ray dominated BL Lacertae objects (HBLs) which includes 150 sources. In this paper, the third of the series, we report the results of a dedicated optical spectroscopy campaign that, together with results from other independent optical follow up programs, led to the spectroscopic identification of all sources in the sample. We carried out a systematic spectroscopic campaign for the observation of all unidentified objects of the sample using the ESO 3.6m, the KPNO 4m, and the TNG optical telescopes. We present new identifications and optical spectra for 76 sources, 50 of which are new BL Lac objects, 18 are sources previously referred as BL Lacs but for which no redshift information was available, and 8 are broad emission lines AGNs. We find that the multi-frequency selection technique used to build the survey is highly efficient (about 90%) in selecting BL Lacs objects. We present positional and spectroscopic information for all confirmed BL Lac objects. Our data allowed us to determined 36 redshifts out of the 50 new BL Lacs and 5 new redshifts for the previously known objects. The redshift distribution of the complete sample is presented and compared with that of other BL Lacs samples. For 26 sources without recognizable absorption features, we calculated lower limits to the redshift using a method based on simulated optical spectra with different ratios between jet and galaxy emission. For a subsample of 38 object with high-quality spectra, we find a correlation between the optical spectral slope, the 1.4 GHz radio luminosity, and the Ca H&K break value, indicating that for powerful/beamed sources the optical light is dominated by the non-thermal emission from the jet.
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