Spectroscopic observations of a sample of 42 flat-spectrum radio sources from the Cosmic Lens All-Sky Survey ( CLASS ) have yielded a mean redshift of < z > = 1.27 with an RMS spread of 0.95 , at a completeness level of 64 \% . The sample consists of sources with a 5-GHz flux density of 25 - 50 mJy , making it the faintest flat-spectrum radio sample for which the redshift distribution has been studied . The spectra , obtained with the Willam Herschel Telescope ( WHT ) , consist mainly of broad-line quasars at z > 1 and narrow-line galaxies at z < 0.5 . Though the mean redshift of flat-spectrum radio sources exhibits little variation over more than two orders of magnitude in radio flux density , there is evidence for a decreasing fraction of quasars at weaker flux levels . In this paper we present the results of our spectroscopic observations , and discuss the implications for constraining cosmological parameters with statistical analyses of the CLASS survey .