We consider the Galerkin finite element approximation of an elliptic Dirichlet boundary control model problem governed by the Laplacian operator. The analytical setting of this problem uses L-2 controls and a "very weak" formulation of the state equation. However, the corresponding finite element approximation uses standard continuous trial and test functions. For this approximation, we derive a priori error estimates of optimal order, which are confirmed by numerical experiments. The proofs employ duality arguments and known results from the L-p error analysis for the finite element Dirichlet projection.