Data from the first physics run at the Relativistic Heavy-Ion Collider at Brookhaven National Laboratory, [Formula presented] collisions at [Formula presented], have been analyzed by the STAR Collaboration using three-pion correlations with charged pions to study whether pions are emitted independently at freeze-out. We have made a high-statistics measurement of the three-pion correlation function and calculated the normalized three-particle correlator to obtain a quantitative measurement of the degree of chaoticity of the pion source. It is fo...