  Neutrino mass explanations of the solar and atmospheric neutrino deficits and
a hot dark matter component require one of three patterns of those masses, as
already pointed out by us. Recently there have been indications of a
non-vanishing amplitude for neutrinoless double beta decay. If this additional
hint of neutrino mass is true, it would make even less likely the one unfavored
pattern (a sterile neutrino giving warm, rather than hot dark matter), would
alter another by making the $\nu_e$ a contributor to the hot dark matter, and
would make the third ($\nu_e$, $\nu_\mu$, and $\nu_\tau$ approximately
degenerate) much more likely than previously. For this third case we construct
a gauge model consistent with other weak interaction data. This model utilizes
a more general version of the see-saw mechanism, which is very likely to be the
source of neutrino mass, if this degenerate pattern is correct. A new supernova
constraint is utilized, and implications and tests of the different mass
matrices are noted.
