We consider the scattering of two spin waves in a uniaxial (easy axis) Heisenberg ferromagnet with single-ion anisotropy. The two-spin-deviation problem is solved exactly at zero temperature. We find (for S>12), in addition to the usual two-spin-wave bound states, a new "single-ion bound state," in which at the zone corner the two spin deviations are on the same site. When the magnitude of the anisotropy is comparable to the exchange interaction, the single-ion bound state becomes the dominant feature of the bound-state spectrum. For arbitrary spin there is a critical anisotropy strength above which the single-ion bound state exists throughout the Brillouin zone. We conclude that the presence of single-ion anisotropy enhances the possibility of experimental observation of the bound states. © 1970 The American Physical Society.