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A technique to transform the waveform of a 14.4 keV photon (time dependence of the photon detection probability or, equivalently, the intensity of the single-photon wave packet) into a regular sequence of short, nearly bandwidth-limited pulses with a controlled number of pulses is proposed. It is based on coherent forward scattering of single X-ray photons from a synchrotron Mössbauer source (SMS) in an optically thick, vibrating, recoilless 57Fe resonant absorber. The possibility of compressing the waveform of an SMS photon into a single short bell-shaped pulse is predicted. The experiment is proposed for compressing a 100 ns duration 14.4 keV single-photon wave packet produced by SMS at the European Synchrotron Radiation Facility (ESRF) into a single bell-shaped pulse of less than 20 ns duration and more than twice the peak intensity. Such single-photon coherent pulses are promising for applications in the fast-developing field of X-ray quantum optics, including possible implementation of quantum memory.