The first observation of an azimuthal cross--section asymmetry with respect to the charge of the incoming lepton beam is reported from a study of hard exclusive electroproduction of real photons. The data have been accumulated by the HERMES experiment at DESY, in which the HERA 27.6 GeV electron or positron beam scattered off an unpolarized hydrogen gas target. The observed asymmetry is attributed to the interference between the Bethe--Heitler process and the Deeply Virtual Compton Scattering (DVCS) process. The interference term is sensitive to DVCS amplitudes, which provide the most direct access to Generalized Parton Distributions.