  The Mathisson-Papapetrou equations in Kerr's background are considered. The
region of existence of highly relativistic planar circular orbits of a spinning
particle in this background and dependence of the particle's Lorentz
$\gamma$-factor on its spin and radial coordinate are investigated. It is shown
that in contrast to the highly relativistic circular orbits of a spinless
particle the corresponding orbits of a spinning particle are allowed in much
wider space region. Some of these orbits show the significant attractive action
of the spin-gravity coupling on a particle and others are caused by the
significant repulsive action. Numerical estimates for electrons, protons and
neutrinos in the gravitational field of black holes are presented.
