  We present a $SU(5)\times U(1)_F$ supersymmetric model for neutrino masses
and mixings that implements the seesaw mechanism by means of the heavy SU(2)
singlets and triplets states contained in three adjoints of SU(5). We discuss
how Abelian $U(1)_F$ symmetries can naturally yield non-hierarchical light
neutrinos even when the heavy states are strongly hierarchical, and how it can
also ensure that $R$--parity arises as an exact accidental symmetry. By
assigning two flavons that break $U(1)_F$ to the adjoint representation of
SU(5) and assuming universality for all the fundamental couplings, the
coefficients of the effective Yukawa and Majorana mass operators become
calculable in terms of group theoretical quantities. There is a single free
parameter in the model, however, at leading order the structure of the light
neutrinos mass matrix is determined in a parameter independent way.
