The photo production of $omega$ mesons on the nuclei C, Ca, Nb and Pb has been measured using the Crystal Barrel/TAPS detector at the ELSA tagged photon facility in Bonn. The dependence of the $omega$ meson cross section on the nuclear mass number has been compared with three different types of models, a Glauber analysis, a BUU analysis of the Giessen theory group and a calculation by the Valencia theory group. In all three cases, the inelastic $omega$ width is found to be $130-150 rm{MeV/c^2}$ at normal nuclear matter density for an average 3-momentum of 1.1 GeV/c. In the restframe of the $omega$ meson, this inelastic $omega$ width corresponds to a reduction of the $omega$ lifetime by a factor $approx 30$. For the first time, the momentum dependent $omega$N cross section has been extracted from the experiment and is in the range of 70 mb.
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