The simplest unified extension of the minimal supersymmetric standard model
with bi-linear R-parity violation provides a predictive scheme for neutrin
o masses which can account for the observed atmospheric and solar neutrino
anomalies in terms of bi-maximal neutrino mixing. The maximality of the atm
ospheric mixing angle arises dynamically, by minimizing the scalar potentia
l, while the solar neutrino problem can be accounted for either by large or
by small mixing oscillations. One neutrino picks up mass by mixing with ne
utralinos, while the degeneracy and masslessness of the other two is lifted
only by loop corrections. Despite the smallness of neutrino masses R-parit
y violation is observable at present and future high-energy colliders, prov
iding an unambiguous cross check of the model.