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(&thgr;1,&thgr;2,&thgr;3)and two squared mass differences. We impose the constraint|m32−m22|≫|m22−m12|to conform with experiment, which requires the&thgr;ito lie nearby one of four 1-parameter domains in &thgr;-space. We exhibit the implications for solar and atmospheric neutrino oscillations in each of these cases. A unique version of the Zeeansatzsurvives confrontation with experimental data, one which necessarily involves maximal just-so vacuum oscillations of solar neutrinos. ©2000 American Institute of Physics.