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Neutrino mass explanations of the solar and atmospheric neutrino deficits anda hot dark matter component require one of three patterns of those masses, asalready pointed out by us. Recently there have been indications of anon-vanishing amplitude for neutrinoless double beta decay. If this additionalhint of neutrino mass is true, it would make even less likely the one unfavoredpattern (a sterile neutrino giving warm, rather than hot dark matter), wouldalter another by making the $\nu_e$ a contributor to the hot dark matter, andwould make the third ($\nu_e$, $\nu_\mu$, and $\nu_\tau$ approximatelydegenerate) much more likely than previously. For this third case we constructa gauge model consistent with other weak interaction data. This model utilizesa more general version of the see-saw mechanism, which is very likely to be thesource of neutrino mass, if this degenerate pattern is correct. A new supernovaconstraint is utilized, and implications and tests of the different massmatrices are noted.Comment: UCSB-HEP-94-03 and UMD-PP-94-90; { Tex File; 14 pages } (File jnl.tex  included and minor misprints corrected.

Accommodating solar and atmospheric neutrino deficits, hot dark matter, and a double β decay signal