  We computed the power spectrum of weak cosmic shear in models with
non-Gaussian primordial density fluctuations. Cosmological initial conditions
deviating from Gaussianity have recently attracted much attention in the
literature, especially with respect to their effect on the formation of
non-linear structures and because of the bounds that they can put on the
inflationary epoch. The fully non-linear matter power spectrum was evaluated
with the use of the physically motivated, semi-analytic halo model, where the
mass function and linear halo bias were suitably corrected for non-Gaussian
cosmologies. In agreement with previous work, we found that a level of
non-Gaussianity compatible with CMB bounds and with positive skewness produces
an increase in power of the order of a few percent at intermediate scales. We
then used the matter power spectrum, together with observationally motivated
background source redshift distributions in order to compute the cosmological
weak lensing power spectrum. We found that the degree of deviation from the
power spectrum of the reference Gaussian model is small compared to the
statistical error expected from even future weak lensing surveys. However,
summing the signal over a large range of multipoles can beat down the noise,
bringing to a significant detection of non-Gaussianity at the level of
$|f_\mathrm{NL}| \simeq $ few tens, when all other cosmological parameters are
held fixed. Finally, we have shown that the constraints on the level of
non-Gaussianity can be improved by $\sim 20%$ with the use of weak lensing
tomography.
