  We propose a model which uses the seesaw mechanism and the lepton number
$\bar L = L_e - L_\mu - L_\tau$ to achieve the neutrino mass spectrum $m_1 =
m_2$ and $m_3 = 0$, together with a lepton mixing matrix $U$ with $U_{e3} = 0$.
In this way, we accommodate atmospheric neutrino oscillations. A small mass
splitting $m_1 > m_2$ is generated by breaking $\bar L$ spontaneously and using
Babu's two-loop mechanism. This allows us to incorporate ``just so''
solar-neutrino oscillations with maximal mixing into the model. The resulting
mass matrix has three parameters only, since $\bar L$ breaking leads
exclusively to a non-zero $ee$ matrix element.
