  We show that a supersymmetric standard model exhibiting anomaly mediated
supersymmetry breaking can generate naturally the observed neutrino mass
spectrum as well mixings when we include bilinear R-parity violation
interactions. In this model, one of the neutrinos gets its mass due to the tree
level mixing with the neutralinos induced by the R-parity violating
interactions while the other two neutrinos acquire their masses due to
radiative corrections. One interesting feature of this scenario is that the
lightest supersymmetric particle is unstable and its decay can be observed at
high energy colliders, providing a falsifiable test of the model.
