  We present a new determination of the parity of the neutral pion via the
double Dalitz decay pi^0 -> e+ e- e+ e-. Our sample, which consists of 30511
candidate decays, was collected from K_L -> pi0 pi0 pi0 decays in flight at the
KTeV-E799 experiment at Fermi National Accelerator Laboratory. We confirm the
negative pi^0 parity, and place a limit on scalar contributions to the pi^0 ->
e+ e- e+ e- decay amplitude of less than 3.3% assuming CPT conservation. The
pi^0 gamma* 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(pi^0 ->
e+ e- e+ e-) = (3.26 +- 0.18) x 10^(-5).
