Data are presented for the reaction pi(-)p-->pi(0)pi(0)n in the range from threshold to p(pi(-))=750 MeV/c. The systematics of the data and multipole analyses are examined for sensitivity to a f(0)(600) ("sigma") meson. A one-pion-exchange mechanism is found to be very weak, or absent. The reaction appears to become dominated by sequential pi(0) decays through the Delta(1232) resonance as the beam momentum increases, along with substantial interference effects from several competing mechanisms.