The fact that neutrinos are massive suggests that the minimal supersymmetric standard model (MSSM) might be extended in order to include three gauge-singlet neutrino superfields with Yukawa couplings of the type $H_2 L ��^c$. We propose to use these superfields to solve the $��$ problem of the MSSM without having to introduce an extra singlet superfield as in the case of the next-to-MSSM (NMSSM). In particular, terms of the type $��^c H_1 H_2$ in the superpotential may carry out this task spontaneously through sneutrino vacuum expectation values. In addition, terms of the type $(��^c)^3$ avoid the presence of axions and generate effective Majorana masses for neutrinos at the electroweak scale. On the other hand, these terms break lepton number and R-parity explicitly implying that the phenomenology of this model is very different from the one of the MSSM or NMSSM. For example, the usual neutralinos are now mixed with the neutrinos. For Dirac masses of the latter of order $10^{-4}$ GeV, eigenvalues reproducing the correct scale of neutrino masses are obtained.

9 pages, latex, title modified. Final version published in PRL