IF a beam of slow neutrons impinges On a solid at grazing incidence, the neutrons reflected can be used to probe the composition and magnetization of the solid near its surface(1,2). In this process, the incident and reflected neutrons generally have identical kinetic energies. Here we report the results of an experiment in which subtle inelastic scattering processes are revealed as relatively large deviations in scattering angle. The neutrons are scattered from a ferromagnetic surface in the presence of a strong ambient magnetic field, and exhibit a small but significant variation in kinetic energy as a function of the reflection angle. This effect is attributable to the Zeeman splitting of the energies of the neutron spin states due to the ambient magnetic field : some neutrons flip their spins upon reflection from the magnetized surface, thereby exchanging kinetic energy for magnetic potential energy. The subtle effects of Zeeman splitting are amplified by the extreme sensitivity of grazing-angle neutron scattering, and might also provide a useful spectroscopic tool if significant practical obstacles (such as low interaction cross-sections) can be overcome.