A Monte Carlo simulation was performed of a “Gedanken Experiment” where ultra-cold neutrons are produced by Bragg reflection from a moving mosaic single crystal. It is shown that ultra-cold neutrons can be obtained by using thermal or cold neutrons (in practice only the latter). The space and velocity distributions of the reflected neutrons are calculated and it is shown that the cross-section of the reflected ultra-cold neutron beam is of elliptical shape. The length of the minor axis of this ellipse has an upper limit determined by the mosaic spread of the moving crystal, and the length of the major axis increases with the ratio of the velocity of the incident neutrons to the velocity of the reflected neutrons. The proposed method of production of ultra-cold neutrons might be useful in cases where a beam of ultra-cold quasi-monochromatic neutrons is required.