We have searched for events in proton-proton collisions, at the CERN Intersecting Storage Rings, which contain a muon and an electron of opposite sign. Such events cannot come from the decay of a single vector meson or from the Drell-Yan process, but can comee from the semileptonic decays of a pair of charmed mesons. We find a signal of 32 ± 16 events. Assumming that the signal is caused by charmed meson pairs, and using a model for the p T and p L distribution of such pairs, we estimate the total inclusive charm production cross section in proton-proton collisions to be (70 ± 36) μb.