  We give analytical arguments and demonstrate numerically the existence of
black hole solutions of the $4D$ Effective Superstring Action in the presence
of Gauss-Bonnet quadratic curvature terms. The solutions possess non-trivial
dilaton hair. The hair, however, is of ``secondary" type", in the sense that
the dilaton charge is expressed in terms of the black hole mass. Our solutions
are not covered by the assumptions of existing proofs of the ``no-hair"
theorem. We also find some alternative solutions with singular metric
behaviour, but finite energy. The absence of naked singularities in this system
is pointed out.
