Two systems of variational equations and inequalities, which characterize the solution of a distributed elliptic control problem governed by the Laplacian with pointwise control and state constraints respectively, are discretized by a mixed finite element method using piecewise linear shape functions. The resulting error estimates of the approximation of the optimal state are derived inH1,2,whereas the corresponding error estimates for the optimal control are given inL2, if the cost functional is strictly coercive, otherwise in negative norms.