  We study the possibility for a measurement of neutrino mass using weak
gravitational lensing. The presence of non-zero mass neutrinos leads to a
suppression of power at small scales and reduces the expected weak lensing
signal. The measurement of such a suppression in the weak lensing power
spectrum allows a direct measurement of the neutrino mass, in contrast to
various other experiments which only allow mass splittings between two neutrino
species. Making reasonable assumptions on the accuracy of cosmological
parameters, we suggest that a weak lensing survey of 100 sqr. degrees can be
easily used to detect neutrinos down to a mass limit of 3.5 eV at the 2 sigma
level. This limit is lower than current limits on neutrino mass, such as from
the Ly-alpha forest and SN1987A. An ultimate weak lensing survey of pi
steradians down to a magnitude limit of 25 can be used to detect neutrinos down
to a mass limit of 0.4 eV at the 2 sigma level, provided that other
cosmological parameters will be known to an accuracy expected from cosmic
microwave background spectrum using the MAP satellite. With improved parameters
estimated from the PLANCK satellite, the limit on neutrino mass from weak
lensing can be further lowered by another factor of 3 to 4. For much smaller
surveys (10 sqr. degrees) that are likely to be first available in the near
future with several wide-field cameras, the presence of neutrinos can be safely
ignored when deriving conventional cosmological parameters such as the mass
density of the Universe. However, armed with cosmological parameter estimates
with other techniques, even such small area surveys allow a strong possibility
to investigate the presence of non-zero mass neutrinos.
