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 $(\theta_1, \theta_2, \theta_3)$ and two squared mass differences. We impose the constraint $|m_3^2 - m_2^2| \gg |m^2_2 - m_1^2|$ to conform with experiment, which requires the $\theta_i$ to lie nearby one of four 1-parameter domains in $\theta$-space. We exhibit the implications for solar and atmospheric neutrino oscillations in each of these cases. A unique version of the Zee {\it ansatz} survives confrontation with experimental data, one which necessarily involves maximal just-so vacuum oscillations of solar neutrinos.