Bose-Einstein Correlations (BEC) of three identical charged pions were studied in 4×1064×106 hadronic Z0Z0 decays recorded with the OPAL detector at LEP. The genuine three-pion correlations, corrected for the Coulomb effect, were separated from the known two-pion correlations by a new subtraction procedure. A significant genuine three-pion BEC enhancement near threshold was observed having an emitter source radius of r3=0.580±0.004 (stat.)±0.029 (syst.) fmr3=0.580±0.004 (stat.)±0.029 (syst.) fm and a strength of λ3=0.504±0.010 (stat.)±0.041 (syst.)λ3=0.504±0.010 (stat.)±0.041 (syst.). The Coulomb correction was found to increase the λ3λ3 value by ∼∼9% and to reduce r3r3 by ∼∼6%. The measured λ3λ3 corresponds to a value of 0.707±0.014 (stat.)±0.078 (syst.)0.707±0.014 (stat.)±0.078 (syst.) when one takes into account the three-pion sample purity. A relation between the two-pion and the three-pion source parameters is discussed.