As a neutron scatters from a target nucleus, there is a small but measurable effect caused by the interaction of the neutron's magnetic dipole moment with that of the partially screened electric field of the nucleus. This spin-orbit interaction is typically referred to as Schwinger scattering and induces a small rotation of the neutron's spin on the order of 10-4 rad for Bragg diffraction from silicon. In our experiment, neutrons undergo greater than 100 successive Bragg reflections from the walls of a slotted, perfect-silicon crystal to amplify the total spin rotation. A magnetic field is employed to insure constructive addition as the neutron undergoes this series of reflections. The strength of the spin-orbit interaction, which is directly proportional to the electric field, was determined by measuring the rotation of the neutron's spin-polarization vector. Our measurements show good agreement with the expected variation of this rotation with the applied magnetic field, while the magnitude of the rotation is ≈40 % larger than expected.