  Density fluctuations in the matter distribution lead to distortions of the
images of distant galaxies through weak gravitational lensing effects. This
provides an efficient probe of the cosmological parameters and of the density
field. In this article, we investigate the statistical properties of the
convergence due to weak gravitational lensing by non-linear structures (i.e. we
consider small angular windows $\theta \la 1'$). Previous studies have shown
how to relate the second and third order moments of the convergence to those of
the density contrast while models based on the Press-Schechter prescription
provide an estimate of the tail of $P(\kappa)$. Here we present a method to
obtain an estimate of the full p.d.f. of the convergence $P(\kappa)$. It is
based on a realistic description of the density field which applies to
overdense as well as underdense regions. We show that our predictions agree
very well with the results of N-body simulations for the convergence. This
could allow one to derive the cosmological parameters
$(\Omega_m,\Omega_{\Lambda})$ as well as the full p.d.f. $P(\delta_R)$ of the
density contrast itself in the non-linear regime from observations. Hence this
gives a very powerfull tool to constrain scenarios of structure formation.
