  We report a measurement of the local acceleration $g$ with ultracold neutrons
based on quantum states in the gravity potential of the Earth. The new method
uses resonant transitions between the states $|1> -> |3>$ and for the first
time between $|1> -> |4>$. The measurements demonstrate that Newton's Inverse
Square Law of Gravity is understood at micron distances at an energy level of
$10^{-14}$ eV with $\frac{\Delta g}{g}=4\times10^{-3}$. The results provide
constraints on any possible gravity-like interaction at a micrometer
interaction range. In particular, a dark energy candidate, the chameleon field
is restricted to $\beta<6.9\times10^{6}$ for $n=2$ (95\% C.L.).
