  The minimal Standard Model extension with the Weinberg operator does
accommodate the observed neutrino masses and mixing, but predicts a
neutrinoless double beta ($0\nu\beta\beta$) decay rate proportional to the
effective electron neutrino mass, which can be then arbitrarily small within
present experimental limits. However, in general $0\nu\beta\beta$ decay can
have an independent origin and be near its present experimental bound; whereas
neutrino masses are generated radiatively, contributing negligibly to
$0\nu\beta\beta$ decay. We provide a realization of this scenario in a simple,
well defined and testable model, with potential LHC effects and calculable
neutrino masses, whose two-loop expression we derive exactly. We also discuss
the connection of this model to others that have appeared in the literature,
and remark on the significant differences that result from various choices of
quantum number assignments and symmetry assumptions. In this type of models
lepton flavor violating rates are also preferred to be relatively large, at the
reach of foreseen experiments. Interestingly enough, in our model this stands
for a large third mixing angle, $\sin^2\theta_{13} \gtrsim 0.008$, when $\mu
\rightarrow eee$ is required to lie below its present experimental limit.
