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 t he 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 (theta(1),theta(2),theta(3)) a nd two squared mass differences. We impose the constraint \m(3)(2) - m(2)(2 )\ >> \m(2)(2) - m(1)(2)\ to conform with experiment, which requires the th eta(i) to lie nearby one of four 1-parameter domains in e-space. We exhibit the implications for solar and atmospheric neutrino oscillations in each o f these cases. A unique version of the Zee ansatz survives confrontation wi th experimental data, one which necessarily involves maximal just-so vacuum oscillations of solar neutrinos. (C) 1999 Published by Elsevier Science B. V. All rights reserved.