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We report on a new experimental approach for the Doppler correction of X-rays emitted by heavy ions, using novel metallic magnetic calorimeter detectors which uniquely combine a high spectral resolution with a broad bandwidth acceptance. The measurement was carried out at the electron cooler of CRYRING@ESR at GSI, Darmstadt, Germany. The X-ray emission associated with the radiative recombination of cooler electrons and stored hydrogen-like uranium ions was investigated using two novel microcalorimeter detectors positioned under 0° and 180° with respect to the ion beam axis. This new experimental setup allowed the investigation of the region of the N, M → L transitions in helium-like uranium with a spectral resolution unmatched by previous studies using conventional semiconductor X-ray detectors. When assuming that the rest-frame energy of at least a few of the recorded transitions is well-known from theory or experiments, a precise measurement of the Doppler shifted line positions in the laboratory system can be used to determine the ion beam velocity using only spectral information. The spectral resolution achievable with microcalorimeter detectors should, for the first time, allow intrinsic Doppler correction to be performed for the precision X-ray spectroscopy of stored heavy ions. A comparison with data from a previous experiment at the ESR electron cooler, as well as the conventional method of conducting Doppler correction using electron cooler parameters, will be discussed.