  We generalize to the case of spinning black holes a recently introduced
``effective one-body'' approach to the general relativistic dynamics of binary
systems. The combination of the effective one-body approach, and of a Pad\'e
definition of some crucial effective radial functions, is shown to define a
dynamics with much improved post-Newtonian convergence properties, even for
black hole separations of the order of $6 GM / c^2$. We discuss the approximate
existence of a two-parameter family of ``spherical orbits'' (with constant
radius), and, of a corresponding one-parameter family of ``last stable
spherical orbits'' (LSSO). These orbits are of special interest for forthcoming
LIGO/VIRGO/GEO gravitational wave observations. It is argued that for most (but
not all) of the parameter space of two spinning holes the effective one-body
approach gives a reliable analytical tool for describing the dynamics of the
last orbits before coalescence. This tool predicts, in a quantitative way, how
certain spin orientations increase the binding energy of the LSSO. This leads
to a detection bias, in LIGO/VIRGO/GEO observations, favouring spinning black
hole systems, and makes it urgent to complete the conservative effective
one-body dynamics given here by adding (resummed) radiation reaction effects,
and by constructing gravitational waveform templates that include spin effects.
Finally, our approach predicts that the spin of the final hole formed by the
coalescence of two arbitrarily spinning holes never approaches extremality.
