  We prove a uniqueness theorem for stationary $D$-dimensional Kaluza-Klein
black holes with $D-2$ Killing fields, generating the symmetry group ${\mathbb
R} \times U(1)^{D-3}$. It is shown that the topology and metric of such black
holes is uniquely determined by the angular momenta and certain other
invariants consisting of a number of real moduli, as well as integer vectors
subject to certain constraints.
