We consider a finite volume scheme with two-point flux approximation (TPFA) to approximate a Laplace problem when the solution exhibits no more regularity than belonging to $H^1_0(\Omega)$. We establish in this case some error bounds for both the solution and the approximation of the gradient component orthogonal to the mesh faces. This estimate is optimal, in the sense that the approximation error has the same order as that of the sum of the interpolation error and a conformity error. A numerical example illustrates the error estimate in the context of a solution with minimal regularity. This result is extended to evolution problems discretized via the implicit Euler scheme in an appendix.