We study error estimates for a finite volume discretization of an elliptic equation. We prove that, for $s\\in [0,1]$, if the exact solution belongs to $H^{1+s}$ and the right-hand side is $f+\\div(G)$ with $f\\in L^2$ and $G\\in (H^s)^N$, then the solution of the finite volume scheme converges in discrete $H^1$-norm to the exact solution, with a rate of convergence of order $h^s$ (where $h$ is the size of the mesh).