A perturbatively nonrenormalizable variant of the Gross-Neveu model of Euclidean quantum field theory with bare propagator p/${\mathrm{p}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\epsilon}}}$ is considered. We outline a rigorous argument proving that by appropriate choice of the bare coupling constant the model may be renormalized nonperturbatively, which results in a theory governed at short distances by a non-Gaussian fixed point of the renormalization group.