  The quenching processes of the thermalized ortho-positronium(o-Ps) on an
oxygen molecule have been studied by the positron annihilation age-momentum
correlation techinique(AMOC). The Doppler broadening spectrum of the 511 keV
gamma-rays from the 2gamma annihilation of o-Ps in O_2 has been measured as a
function of the o-Ps age. The rate of the quenching, consisting of the pickoff
and the spin-conversion, is estimated from the positron lifetime spectrum. The
ratio of the pickoff quenching rate to the spin-conversion rate is deduced from
the Doppler broadening of the 511 keV gamma-rays from the annihilation of the
o-Ps. The pickoff parameter ^1Z_eff, the effective number of the electrons per
molecule which contribute to the pickoff quenching, for O_2 is determined to be
0.6 +- 0.4. The cross-section for the elastic spin-conversion quenching is
determined to be (1.16 +- 0.01) * 10^{-19} cm^2.
