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Abstract. Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants ([10]), stable on deforma-tions. From this factorization, one can compute the fundamental group of the complement of the branch curve, either in C2 or in CP2. In this article, we show that these groups, for the Hirzebruch surface F1,(a,b), are almost-solvable. That is- they are an extension of a solvable group, which strengthen the conjecture on degeneratable surfaces (see [13]). keywords:Hirzebruch surfaces, degeneration, generic projection, branch curve, braid mon-odromy, fundamental group, classification of surfaces