The helicity dependence of the total cross section for the (gamma) over right arrow(p) over right arrow -> p pi(0)pi(0) reaction has been measured for the first time at incident photon energies from 400 to 800 MeV. The measurement, performed at the tagged photon beam facility of the MAMI accelerator in Mainz, used the large acceptance detector DAPHNE and a longitudinally polarized frozen-spin target. This channel is found to be excited predominantly when the photon and proton have a parallel spin orientation, most likely due to the intermediate production of the D-13(1520) resonance. However, the contribution of the antiparallel spin configuration, arising from other reaction mechanisms, is also not negligible. This result gives important new information to resolve the existing model discrepancies in the identification of the nucleon resonances contributing to this channel.