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.
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