We propose a new mechanism of exciton-magnon absorption in the 1D Heisenberg antiferromagnet KCuF3. It yields the correct description of the relative intensity of the spectral lines in both polarizations E c and E‖c. The magnon density of states has been calculated. The origin of the fine sideband structure in KCuF3 can be explained. The sharp magnon lines, observed in Ref. , correspond to the low energy singularities in the magnon density of states.