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We prove that a very ample special line bundle $L$ of degree $d>\left(3g-1\right)/2$ on a general $k$-gonal curve is normally generated if the degree of the base locus of its dual bundle $K{L}^{-1}$ does not exceed $c\left(k-2\right)/2$, where $c:=d-\left(3g-1\right)/2$.