  [ABRIDGED] We present the discovery and follow-up observations of 142 pulsars
found in the Parkes 20-cm multibeam pulsar survey of the Galactic plane. These
new discoveries bring the total number of pulsars found by the survey to 742.
In addition to tabulating spin and astrometric parameters, along with pulse
width and flux density information, we present orbital characteristics for 13
binary pulsars which form part of the new sample. Combining these results from
another recent Parkes multibeam survey at high Galactic latitudes, we have a
sample of 1008 normal pulsars which we use to carry out a determination of
their Galactic distribution and birth rate. We infer a total Galactic
population of 30000 +/- 1100 potentially detectable pulsars (i.e. those beaming
towards us) having 1.4-GHz luminosities above 0.1 mJy kpc squared. Using a
pulsar current analysis, we derive the birth rate of this population to be 1.4
+/- 0.2 pulsars per century. An important conclusion from our work is that the
inferred radial density function of pulsars depends strongly on the assumed
distribution of free electrons in the Galaxy. As a result, any analyses using
the most recent electron model of Cordes & Lazio predict a dearth of pulsars in
the inner Galaxy. We show that this model can also bias the inferred pulsar
scale height with respect to the Galactic plane. Combining our results with
other Parkes multibeam surveys we find that the population is best described by
an exponential distribution with a scale height of 330 pc.
