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Because of their long lifetimes, the omega and phi 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 H-2, 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 omega meson is stronger than that reported by the CBELSA-TAPS experiment and cannot be explained by recent theoretical models.