  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.
