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Pinning down the total neutrino mass and the dark energy equation of state isa key aim for upcoming galaxy surveys. Weak lensing is a unique probe of thetotal matter distribution whose non-Gaussian statistics can be quantified bythe one-point probability distribution function (PDF) of the lensingconvergence. We calculate the convergence PDF on mildly non-linear scales fromfirst principles using large-deviation statistics, accounting for dark energyand the total neutrino mass. For the first time, we comprehensively validatethe cosmology-dependence of the convergence PDF model against large suites ofsimulated lensing maps, demonstrating its percent-level precision and accuracy.We show that fast simulation codes can provide highly accurate covariancematrices, which can be combined with the theoretical PDF model to performforecasts and eliminate the need for relying on expensive N-body simulations.Our theoretical model allows us to perform the first forecast for theconvergence PDF that varies the full set of $\Lambda$CDM parameters. Our Fisherforecasts establish that the constraining power of the convergence PDF comparesfavourably to the two-point correlation function for a Euclid-like survey areaat a single source redshift. When combined with a CMB prior from Planck, thePDF constrains both the neutrino mass $M_\nu$ and the dark energy equation ofstate $w_0$ more strongly than the two-point correlation function.

Nuw CDM cosmology from the weak lensing convergence PDF