  We present a theoretical study of the motion of the antihydrogen atom
($\bar{H}$) in the Earth's gravitational field above a material surface. We
predict that $\bar{H}$ atom, falling in the Earth's gravitational field above a
material surface, would settle in long-living quantum states. We point out a
method of measuring the difference in energy of $\bar{H}$ in such states that
allow us to apply spectroscopy of gravitational levels based on
atom-interferometric principles. We analyze a general feasibility to perform
experiments of this kind. We point out that such experiments provide a method
of measuring the gravitational force ($Mg$) acting on $\bar{H}$ and they might
be of interest in a context of testing the Weak Equivalence Principle for
antimatter.
