A 1D model of a finite-length spin chain with the Dzyaloshinskii–Moriya interaction is considered. An algorithm for simulation of the problem is developed, and calculations that demonstrate the existence of discrete magnetic breathers of various types in the phase of forced ferromagnetism of a chiral helimagnet are presented. It is shown that the analysis with allowance for the Dzyaloshinskii–Moriya interaction yields a decrease in the amplitude of discrete magnetic breathers.