We prove the existence of a smooth and non-degenerate curve X ⊂ P n , n ≥ 8, with deg(X) = d, pa(X) = g, h 1 (NX (−1)) = 0 and general moduli for all (d, g, n) such that d ≥ (n − 3) g/2 + n + 3. It was proved by Ch. Walter that for n ≥ 4 the inequality 2d ≥ (n − 3)g + 4 is a necessary condition for the existence of a curve with h 1 (NX (−1)) = 0. 2010 Mathematics Subject Classification. 14H50.