  We consider an effective Majorana neutrino mass operator with the
Friedberg-Lee symmetry; i.e., it is invariant under the transformation
\nu_\alpha -> \nu_\alpha + z (for \alpha = e, \mu, \tau) with z being a
space-time independent constant element of the Grassmann algebra. We show that
this new flavor symmetry can be broken in such a nontrivial way that the
lightest neutrino remains massless but an experimentally-favored neutrino
mixing pattern is achievable. In particular, we get a novel prediction for the
unknown neutrino mixing angle \theta_13 in terms of two known angles:
\sin\theta_13 = \tan\theta_12 |(1- \tan\theta_23)/ (1+\tan\theta_23)|. The
model can simply be generalized to accommodate CP violation and be combined
with the seesaw mechanism.
