  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.
