  We examine the evolution, under gravitational radiation reaction, of slightly
eccentric equatorial orbits of point particles around Kerr black holes. Our
method involves numerical integration of the Sasaki-Nakamura equation. It is
discovered that such orbits decrease in eccentricity throughout most of the
inspiral, until shortly before the innermost stable circular orbit (ISCO), when
a critical radius $r_{\text{crit}}$ is reached beyond which the inspiralling
orbits increase in eccentricity. It is shown that the number of orbits
remaining in this last (eccentricity increasing) phase of the inspiral is an
order of magnitude less for prograde orbits around rapidly spinning black holes
than for retrograde orbits. In the extreme limit of a Kerr black hole with spin
parameter $a=1$, this critical radius descends into the ``throat'' of the black
hole.
