  We study the geodesic equation in the space-time of a Kerr black hole pierced
by an infinitely thin cosmic string and give the complete set of analytical
solutions of this equation for massive and massless particles in terms of Mino
time that allows to decouple the r- and theta-component of the geodesic
equation. The solutions of the geodesic equation can be classified according to
the particle's energy and angular momentum, the mass and angular momentum per
mass of the black hole. We give examples of orbits showing the influence of the
cosmic string. We also discuss the perihelion shift and the Lense-Thirring
effect for bound orbits and show that the presence of a cosmic string enhances
both effects. Comparing our results with experimental data from the LAGEOS
satellites we find an upper bound on the energy per unit length of a string
piercing the earth which is approximately 10^{16} kg/m. Our work has also
applications to the recently suggested explanation of the alignment of the
polarization vector of quasars using remnants of cosmic string decay in the
form of primordial magnetic field loops.
