  We present spectra for a sample of radio sources from the FIRST survey, and
use them to define the form of the redshift distribution of radio sources at
mJy levels.We targeted 365 sources and obtained 46 redshifts (13 per cent of
the sample). We find that our sample is complete in redshift measurement to R
$\sim 18.6$, corresponding to $z\sim 0.2$. Early-type galaxies represent the
largest subset (45 per cent) of the sample and have redshifts $0.15\la z \la
0.5$ ; late-type galaxies make up 15 per cent of the sample and have redshifts
$0.05\la z \la 0.2$; starbursting galaxies are a small fraction ($\sim 6$ per
cent), and are very nearby ($z\la 0.05$). Some 9 per cent of the population
have Seyfert1/quasar-type spectra, all at $z\ga 0.8$, and there are 4 per cent
are Seyfert2 type galaxies at intermediate redshifts ($z\sim 0.2$). Using our
measurements and data from the Phoenix survey, we obtain an estimate for $N(z)$
at $S_{1.4 \rm {GHz}}\ge 1$ mJy and compare this with model predictions. At
variance with previous conclusions, we find that the population of starbursting
objects makes up $\la 5 $ per cent of the radio population at S $\sim 1$ mJy.
