  We present phase shift measurements for neutron matter waves in vacuum and in
low pressure Helium using a method originally developed for neutron scattering
length measurements in neutron interferometry. We search for phase shifts
associated with a coupling to scalar fields. We set stringent limits for a
scalar chameleon field, a prominent quintessence dark energy candidate. We find
that the coupling constant $\beta$ is less than 1.9 $\times10^7$~for $n=1$ at
95\% confidence level, where $n$ is an input parameter of the self--interaction
of the chameleon field $\varphi$ inversely proportional to $\varphi^n$.
