Beam-helicity asymmetries for the two-pion-photoproduction reaction gamma + p --> p pi+ pi- have been studied for the first time in the resonance region for center-of-mass energies between 1.35 GeV and 2.30 GeV. The experiment was performed at Jefferson Lab with the CEBAF Large Acceptance Spectrometer using circularly polarized tagged photons incident on an unpolarized hydrogen target. Beam-helicity-dependent angular distributions of the final-state particles were measured. The large cross-section asymmetries exhibit strong sensitivity to the kinematics and dynamics of the reaction. The data are compared with the results of various phenomenological model calculations, and show that these models currently do not provide an adequate description for the behavior of this new observable.