  Nonminimal spin-gravity interaction through unit gravimagnetic moment leads
to modified Mathisson-Papapetrou-Tulczyjew-Dixon equations with improved
behavior in the ultrarelativistic limit. We present exact Hamiltonian of the
resulting theory and compute an effective $\frac{1}{c^2}$\,-Hamiltonian and
leading post-Newtonian corrections to the trajectory and spin. Gravimagnetic
moment causes the same precession of spin ${\bf S}$ as a fictitious rotation of
the central body with angular momentum ${\bf J}=\frac{M}{m}{\bf S}$. So the
modified equations imply a number of qualitatively new effects, that could be
used to test experimentally, whether a rotating body in general relativity has
null or unit gravimagnetic moment.
