  This paper is part of a series reporting the results of the ATESP radio
survey obtained at 1.4 GHz with the Australia Telescope Compact Array. The
survey consists of 16 radio mosaics with ~8"x14" resolution and uniform
sensitivity (1sigma noise level ~79 microJy) over the whole area of the ESO
Slice Project redshift survey (~26 sq. degrees at decl. = -40 degrees). The
ATESP survey has produced a catalogue of 2960 radio sources down to a flux
limit (6sigma) of ~0.5 mJy. In this paper we present the 1.4 GHz log N - log S
relation derived from the ATESP radio source catalogue. The possible causes of
incompleteness at the faint end of the source counts are extensively discussed
and their effects are quantified and corrected for. The ATESP counts are
consistent with others reported in the literature, even though some significant
discrepancies are present at low fluxes (below a few mJy). We investigate
whether such discrepancies may be explained in terms of field-to-field
anisotropies, considering the fact that all the existing sub-mJy surveys cover
small areas of sky (from a fraction of square degree to a few square degrees).
We stress that the ATESP survey, covering 26 sq. degrees, provides the best
determination of source counts available today in the flux range 0.7 <
S(1.4GHz) < 2 mJy.
