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We present a new determination of the parity of the neutral pion via the double Dalitz decay pi0-->e+e-e+e-. Our sample, which consists of 30,511 candidate decays, was collected from KL-->pi0pi0pi0 decays in flight at the KTeV-E799 experiment at Fermi National Accelerator Laboratory. We confirm the negative pi0 parity and place a limit on scalar contributions to the pi0-->e+e-e+e- decay amplitude of less than 3.3% assuming CPT conservation. The pi0gamma*gamma* form factor is well described by a momentum-dependent model with a slope parameter fit to the final state phase-space distribution. Additionally, we have measured the branching ratio of this mode to be B(pi0-->e+e-e+e-)=(3.26+/-0.18)x10(-5).