Can the Zee ansatz for neutrino masses be correct?

Working in the framework of three chiral neutrinos with Majorana masses, we investigate a scenario first realized in an explicit model by Zee: that the neutrino mass matrix is strictly off-diagonal in the flavor basis, with all its diagonal entries precisely zero. This CP-conserving ansatz leads to two relations among the three mixing angles (θ1,θ2,θ3) and two squared mass differences. We impose the constraint |m32−m22|≫|m22−m12| to conform with experiment, which requires the θi to lie nearby one of four 1-parameter domains in θ-space. We exhibit the implications for solar and atmospheric neutrino oscillations in each of these cases. A unique version of the Zee ansatz survives confrontation with experimental data, one which necessarily involves maximal just-so vacuum oscillations of solar neutrinos.