ABSTRACT: 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 mvtree ~~ (tetha2)& eV where tetha ~_ 0.22 is the U(1)H breaking parameter and & is an integer number that depends on the horizontal charges of the leptons. For &=1 a unique self-consistent model arises in which (i) all the superpotential trilinear R-parity violating couplings are forbidden by holomorphy; (ii) mvtree falls in the range suggested by the atmospheric neutrino problem; (iii) radiative contributions to neutrino masses are strongly suppressed resulting in m2 solar ~~ fe 10^8 eV2 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 tetha and can naturally be large.