The minimal Standard Model extension with the Weinberg operator does accommodate the observed neutrino masses and mixing, but predicts a neutrinoless double beta (0νββ) decay rate proportional to the effective electron neutrino mass, which can be then arbitrarily small within present experimental limits. However, in general 0νββ decay can have an independent origin and be near its present experimental bound; whereas neutrino masses are generated radiatively, contributing negligibly to 0νββ 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, sin2θ13>∼0.008, when μ→eee is required to lie below its present experimental limit.