Zee Mass Matrix and Bi-Maximal Neutrino Mixing

We investigate neutrino masses and mixings within the framework of the Zee mass matrix, with three lepton flavors. It is shown that the bi-maximal solution is the only possibility to reconcile atmospheric and solar neutrino data, within this ansatz. We obtain two almost degenerate neutrinos, which are mixtures of all three neutrino flavors, with heavy masses $\simeq \sqrt{\Delta m_{atm}^2}$. The predicted mass of the lightest neutrino, which should consist mostly of $\nu_{\mu}$ and $\nu_{\tau}$, is $\simeq \Delta m_{\odot}^2/(2\sqrt{\Delta m_{atm}^2})$.