Absorption spectra of the one-dimensional Ising-like antiferromagnet CsFeCl3·2H2O were investigated in steady magnetic fields up to 130 kOe. Sublattice splittings were observed in an exciton line and its magnon sideband. The branch of the magnon sideband and its original exciton line are crossed at approximately 75 kOe. In the higher field the energy of the magnon sideband becomes lower than that of the exciton line. This unusual behavior was explained in terms of the exciton-magnon interaction and magnetic Franck-Condon effect.