  In [arXiv:hep-th/0611083] we announced the complete integrability of geodesic
motion in the general higher-dimensional rotating black-hole spacetimes. In the
present paper we prove all the necessary steps leading to this conclusion. In
particular, we demonstrate the independence of the constants of motion and the
fact that they Poisson commute. The relation to a different set of constants of
motion constructed in [arXiv:hep-th/0612029] is also briefly discussed.
