  Experiments in progress may confirm a nonzero neutrinoless double beta decay
rate in conflict with the cosmological upper limit on neutrino masses and thus
require new physics beyond the Standard Model. A natural candidate is the
Left-Right symmetric theory, which led originally to neutrino mass and the
seesaw mechanism. In the absence of cancelations of large Dirac Yukawa
couplings, we show how such a scenario would require a low scale of Left-Right
symmetry breaking roughly below 10 TeV, tantalizingly close to the LHC reach.
