  The simplest unified extension of the MSSM with bi-linear R--Parity violation
naturally predicts a hierarchical neutrino mass spectrum, in which one neutrino
acquires mass by mixing with neutralinos, while the other two get mass
radiatively. We have performed a full one-loop calculation of the
neutralino-neutrino mass matrix in the bi-linear \rp MSSM, taking special care
to achieve a manifestly gauge invariant calculation. Moreover we have performed
the renormalization of the heaviest neutrino, needed in order to get meaningful
results. The atmospheric mass scale and maximal mixing angle arise from
tree-level physics, while solar neutrino scale and oscillations follow from
calculable one-loop corrections. If universal supergravity assumptions are made
on the soft-supersymmetry breaking terms then the atmospheric scale is
calculable as a function of a single \rp violating parameter by the
renormalization group evolution due to the non-zero bottom quark Yukawa
coupling. The solar neutrino problem must be accounted for by the small mixing
angle (SMA) MSW solution. If these assumptions are relaxed then one can
implement large mixing angle solutions, either MSW or just-so. The theory
predicts the lightest supersymmetic particle (LSP) decay to be observable at
high-energy colliders, despite the smallness of neutrino masses indicated by
experiment. This provides an independent way to test this solution of the
atmospheric and solar neutrino anomalies.
