  Gravitomagnetic effects are characterized by two phenomena: first, the
geodetic effect which describes the precession of the spin of a gyroscope in a
free orbit around a massive object, second, the Lense-Thirring effect which
describes the precession of the orbital plane about a rotating source mass. We
calculate both these effects in the fourth-order theory of conformal Weyl
gravity for the test case of circular orbits. We show that for the geodetic
effect a linear term arises which may be interesting for high radial orbits,
whereas for the Lense-Thirring effect the additional term has a diminishing
effect for most orbits. Circular orbits are also considered in general leading
up to a generalization of Kepler's third law.
