Curves of genus g whose canonical model lies on a surface of degree g+1

Let $C$ be a non--hyperelliptic curve of genus $g$. We prove that, if the minimal degree of a surface containing the canonical model of $C$ in $\dualp{g-1}$ is $g+1$, then either $g\ge9$ and $C$ carries exactly one $g^1_4$, or $7\le g\le15$ and $C$ is birationally isomorphic to a plane septic with at most double points as singularities.