  A test particle of mass mu on a bound geodesic of a Kerr black hole of mass M
>> mu will slowly inspiral as gravitational radiation extracts energy and
angular momentum from its orbit. This inspiral can be considered adiabatic when
the orbital period is much shorter than the timescale on which energy is
radiated, and quasi-circular when the radial velocity is much less than the
azimuthal velocity. Although the inspiral always remains adiabatic provided mu
<< M, the quasi-circular approximation breaks down as the particle approaches
the innermost stable circular orbit (ISCO). In this paper, we relax the
quasi-circular approximation and solve the radial equation of motion explicitly
near the ISCO. We use the requirement that the test particle's 4-velocity
remain properly normalized to calculate a new contribution to the difference
between its energy and angular momentum. This difference determines how a black
hole's spin changes following a test-particle merger, and can be extrapolated
to help predict the mass and spin of the final black hole produced in
finite-mass-ratio black-hole mergers. Our new contribution is particularly
important for nearly maximally spinning black holes, as it can affect whether a
merger produces a naked singularity.
