Strong ultraviolet resonance transitions are observed routinely both in the Galactic interstellar medium and in quasar absorption systems. The quality of the astronomical spectroscopic data now available demands more precise laboratory rest wavelengths. Of particular interest is the accuracy with which one can constrain space-time variations in fundamental constants using quasar spectra. A recent analysis by Webb et al. of 25 quasar spectra using Mg and Fe transitions tentatively suggests that the fine-structure constant was smaller at earlier epochs. To permit a check on this result, and to allow further more extensive investigations, we have carried out a new determination of the laboratory wavelengths of Mg i 2853 Å, Mg ii 2796 Å and Mg ii 2803 Å by high-resolution Fourier transform spectroscopy. Our results for Mg ii 2796 Å are consistent with the value measured independently by two other groups. To our knowledge, no previous measurements of comparable precision exist for Mg i 2853 Å and Mg ii 2803 Å.