Solid phase epitaxy of Er-implanted amorphous Si results in segregation and trapping of the Er, incorporating up to 2×1020 Er/cm3 in single-crystal Si. Segregation occurs despite an extremely low Er diffusivity in bulk amorphous Si of ≤10-17 cm2/s, and the narrow segregation spike (measured width ≊3 nm) suggests that kinetic trapping is responsible for the nonequilibrium concentrations of Er. The dependence of trapping on temperature, concentration, and impurities indicates instead that thermodynamics controls the segregation. We propose that Er, in analogy to transition metals, diffuses interstitially in amorphous Si, but is strongly bound at trapping centers. The binding enthalpy of these trapping sites causes the amorphous phase to be energetically favorable for Er, so that at low concentrations the Er is nearly completely segregated. Once the concentration of Er in the segregation spike exceeds the amorphous trap center concentration, though, more Er is trapped in the crystal. We also observe similar segregation and trapping behavior for another rare-earth element, Pr.