  We present a general method for the reconstruction of the one-point
Probability Distribution Function of the local aperture mass in weak lensing
maps. Exact results, that neglect the lens-lens coupling and departure form the
Born approximation, are derived for both the quasilinear regime at leading
order and the strongly nonlinear regime assuming the tree hierarchical model is
valid. We describe in details the projection effects on the properties of the
PDF and the associated generating functions. In particular, we show how the
generic features which are common to both the quasilinear and nonlinear regimes
lead to two exponential tails for $P(\Map)$. We briefly investigate the
dependence of the PDF with cosmology and with the shape of the angular filter.
Our predictions are seen to agree reasonably well with the results of numerical
simulations and should be able to serve as foundations for alternative methods
to measure the cosmological parameters that take advantage of the full shape of
the PDF.
