Information on hadron properties in the nuclear medium has been derived from the photoproduction of omega mesons on the nuclei C, Ca, Nb, and Pb 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 Boltzmann-Uehling-Uhlenbeck 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 MeV/c(2) at normal nuclear matter density for an average 3-momentum of 1.1 GeV/c. In the rest frame of the omega meson, this inelastic omega width corresponds to a reduction of the omega lifetime by a factor approximately 30. For the first time, the momentum dependent omegaN cross section has been extracted from the experiment and is in the range of 70 mb.