  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 Lagrangian multiplier. Our final formulation fulfills
the Babu\v{s}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.
Numerical experiments confirm our theoretical findings.
