We present a method for the numerical approximation of distributed optimal control problems constrained by parabolic partial differential equations. We complement the first-order optimality condition by a recently developed space-time variational formulation of parabolic equations which is coercive in the energy norm, and a Lagrange multiplier. Our final formulation fulfills the Babu & scaron;ka-Brezzi conditions on the continuous as well as discrete level, without restrictions. Consequently, we can allow for final-time desired states, and obtain an a posteriori error estimator which is efficient and reliable up to an additional discretization error of the adjoint problem. Numerical experiments confirm our theoretical findings.