  The Goos-H\"anchen (GH) shift describes a phenomenon in which a
specularly-reflected beam is laterally translated along the reflecting surface
such that the incident and reflected rays no longer intersect at the surface.
Using a neutron spin-echo technique and a specially-designed magnetic
multilayer mirror, we have measured the relative phase between the reflected up
and down neutron spin states in total reflection. The relative GH shift
calculated from this phase shows a strong resonant enhancement at a particular
incident neutron wavevector, which is due to a waveguiding effect in one of the
magnetic layers. Calculations based on the observed phase difference between
the neutron states indicate a propagation distance along the waveguide layer of
0.65 mm for the spin-down state, which we identify with the magnitude of the
giant GH shift. The existence of a physical GH shift is confirmed by the
observation of neutron absorption in the waveguide layer. We propose ways in
which our experimental method may be exploited for neutron quantum-enhanced
sensing of thin magnetic layers.
