Mass measurements of short-lived uranium projectile fragments were performed for the first time with a Multiple-Reflexion-Time-of-Flight Mass Spectrometer (MR-TOF-MS). A major part of this doctoral work was a novel development of a data analysis method for the MR-TOF-MS mass measurements of exotic nuclei at the fragment separator FRS at GSI. The developed method was successfully applied to the data obtained from two pilot experiments with the MR-TOF-MS at the FRS in 2012 and 2014. A substantial upgrade of the experimental setup of the MR-TOF-MS was also performed in the frame work of this doctoral thesis after the first run.In the experiments projectile fragments were created with 1000 MeV/u 238U ions in a Be/Nb target at the entrance of the in-flight separator FRS. The exotic nuclei were spatially separated, energy bunched and slowed down with the ion-optical system of the FRS combined with monoenergetic and homogeneous degraders. At the final focal plane of the FRS the fragments were completely slowed down and thermalized in a cryogenic stopping cell (CSC) filled with 3-5 mg/cm^2 pure helium gas. The exotic nuclei were fast extracted from the CSC to enable mass measurements of very short-lived fragments with the MR-TOF-MS. The achievement of this goal was successfully demonstrated with the mass measurement of 220Ra ions with a half-life of 17.9 ms and 11 detected events. The mass measurements of the isobars 211Fr, 211Po and 211Rn have clearly demonstrated the scientific potential of the MR-TOF-MS for the investigation of exotic nuclei and the power of the data analysis system. Difficult measurements with overlapping mass distributions with only a few counts in the measured spectra were the challenge for the new data analysis method based on the maximum likelihood method.The drifts during the measurements were corrected with the developed time-resolved calibration method. After the improvements of the setup as a consequence of the experience of the first experiment in 2012 and the applied time-resolved calibration method a mass resolving power of 400,000 has been achieved in the experiment in 2014. The achieved mass accuracy in these pilot experiments were about $mathsf{1cdot 10^{-6}}$. The contribution of the software and the resulting systematic errors were in the 10$^{-8}$ range. The reliability of the present analysis method was carefully checked in detailed simulations with a realistic peak shape approximated by an exponentially modified Gaussian distribution. Both list mode data and measured histograms were treated in the data analysis. The analysis method was tested with strongly overlapping mass distributions and low count rates including a variable amount of back ground. In summary, the experimental setup for mass measurements of very rare and short-lived nuclei and the corresponding data analysis have reached with this work and results of the present thesis a great potential for high-resolution measurements in future experiments. There mass measurements with 10 events can be performed with a residual uncertainty of $mathsf{4.5 cdot 10^{-7}}$ at a mass resolving power of 400,000.
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