  Recently, quantum states of ultra-cold neutrons in the Earth's gravitational
field have been observed for the first time. From the fact that they are
consistent with Newtonian gravity on the 10 %-level, analytical limits on alpha
and lambda of short-range Yukawa-like additional interactions are derived
between lambda = 1 micrometer and 1 mm. We arrive for lambda > 10 micrometer at
alpha < 2 \cdot 10^11 at 90 % confidence level. This translates into a limit
g_s g_p / (\hbar c) < 2 \cdot 10^{-15} on the pseudo-scalar coupling of axions
in the previously experimentally unaccessible astrophysical axion window.
