Let X be a smooth proper complex variety. We consider the cycle map from the Chow ring to the ring of the Deligne cohomology. If this cycle map is injective (modulo torsion), then it has to be surjective as well, and the groups $\mathrm {H}^p(\mathrm {X}, \mathcal {K}_{p+1})$ are generated by constant functions on codimension $p$ cycles (modulo torsion). This generalizes Jannsen's results concerning the cycle map with values in the Betti cohomology.