  We present the first analytical computation of the (conservative)
gravitational self-force correction to the periastron advance around a spinning
black hole. Our result is accurate to the second order in the rotational
parameter and through the 9.5 post-Newtonian level. It has been obtained as the
circular limit of the correction to the gyroscope precession invariant along
slightly eccentric equatorial orbits in the Kerr spacetime. The latter result
is also new and we anticipate here the first few terms only of the
corresponding post-Newtonian expansion.
