Because of their long lifetimes, the ω and ϕ mesons are the ideal candidates for the study of possible modifications of the in-medium meson-nucleon interaction through their absorption inside the nucleus. During the E01-112 experiment at the Thomas Jefferson National Accelerator Facility, the mesons were photoproduced from 2H, C, Ti, Fe, and Pb targets. This Letter reports the first measurement of the ratio of nuclear transparencies for the e+e− channel. The ratios indicate larger in-medium widths compared with what have been reported in other reaction channels. The absorption of the ω meson is stronger than that reported by the CBELSA-TAPS experiment and cannot be explained by recent theoretical models.