  We derive the X-ray luminosity function (XLF) of the X-ray source population
detected in the Chandra observation of NGC4038/39 (the Antennae).
  We explicitly include photon counting and spectral parameter uncertainties in
our calculations. The cumulative XLF is well represented by a flat power law
($\alpha=-0.47$), similar to those describing the XLFs of other star-forming
systems (e.g. M82, the disk of M81), but different from those of early type
galaxies. This result associates the X-ray source population in the Antennae
with young High Mass X-ray Binaries. In comparison with less actively
star-forming galaxies, the XLF of the Antennae has a highly significant excess
of sources with luminosities above 10^{39} erg\s (Ultra Luminous Sources;
ULXs). We discuss the nature of these sources, based on the XLF and on their
general spectral properties, as well as their optical counterparts discussed in
Paper III. We conclude that the majority of the ULXs cannot be intermediate
mass black-holes (M > 10-1000 \msun) binaries, unless they are linked to the
remnants of massive Population III stars (the Madau & Rees model). Instead,
their spatial and multiwavelength properties can be well explained by beamed
emission as a consequence of supercritical accretion.
  Binaries with a neutron star or moderate mass black-hole (up to 20\msun), and
B2 to A type star companions would be consistent with our data. In the beaming
scenario, the XLF should exibit caracteristic breaks that will be visible in
future deeper observations of the Antennae.
