  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.
