  We study extensions of supersymmetric models without R-parity which include
an anomalous U(1)_H horizontal symmetry. Bilinear R-parity violating terms
induce a neutrino mass at tree level of approximately $(\theta^2)^\delta$ eV
where $\theta\approx 0.22$ is the U(1)_H breaking parameter and $\delta$ is an
integer number that depends on the horizontal charges of the leptons. For
$\delta=1$ a unique self-consistent model arises in which i) all the
superpotential trilinear R-parity violating couplings are forbidden by
holomorphy; ii) the tree level neutrino mass falls in the range suggested by
the atmospheric neutrino problem; iii) radiative contributions to neutrino
masses are strongly suppressed resulting in a squared solar mass difference of
few 10^{-8} eV^2 which only allows for the LOW (or quasi-vacuum) solution to
the solar neutrino problem; iv) the neutrino mixing angles are not suppressed
by powers of $\theta$ and can naturally be large.
