  The energy eigenstates of a spin$-\frac{1}{2}$ particle in a magnetic field
confined to a plane, define a planar spin. If the particle moves adiabatically
around a loop in this plane, it picks up a topological Berry phase that can
only be an integer multiple of $\pi$. We propose a neutron interferometry test
of the Berry phase for a circulating planar spin induced by a magnetic field
caused by a very long current-carrying straight wire perpendicular to the
plane. This Berry phase causes destructive interference in the direction of the
incoming beam of thermal neutrons moving through a triple-Laue interferometer.
