  We present a non-canonically symplectic integration scheme tailored to
numerically computing the post-Newtonian motion of a spinning black-hole
binary. Using a splitting approach we combine the flows of orbital and spin
contributions. In the context of the splitting, it is possible to integrate the
individual terms of the spin-orbit and spin-spin Hamiltonians analytically,
exploiting the special structure of the underlying equations of motion. The
outcome is a symplectic, time-reversible integrator, which can be raised to
arbitrary order by composition. A fourth-order version is shown to give
excellent behavior concerning error growth and conservation of energy and
angular momentum in long-term simulations. Favorable properties of the
integrator are retained in the presence of weak dissipative forces due to
radiation damping in the full post-Newtonian equations.
