  Large future galaxy cluster surveys, combined with cosmic microwave
background observations, can achieve a high sensitivity to the masses of
cosmologically important neutrinos. We show that a weak lensing selected sample
of ~100,000 clusters could tighten the current upper bound on the sum of masses
of neutrino species by an order of magnitude, to a level of 0.03 eV. Since this
statistical sensitivity is below the best existing lower limit on the mass of
at least one neutrino species, a future detection is likely, provided that
systematic errors can be controlled to a similar level.
