  We apply standard post-Newtonian methods in general relativity to locate the
innermost circular orbit (ICO) of irrotational and corotational binary
black-hole systems. We find that the post-Newtonian series converges well when
the two masses are comparable. We argue that the result for the ICO which is
predicted by the third post-Newtonian (3PN) approximation is likely to be very
close to the ``exact'' solution, within 1% of fractional accuracy or better.
The 3PN result is also in remarkable agreement with a numerical calculation of
the ICO in the case of two corotating black holes moving on exactly circular
orbits. The behaviour of the post-Newtonian series suggests that the
gravitational dynamics of two bodies of comparable masses does not resemble
that of a test particle on a Schwarzschild background. This leads us to
question the validity of some post-Newtonian resummation techniques that are
based on the idea that the field generated by two black holes is a deformation
of the Schwarzschild space-time.
