We have used diode laser absorption spectroscopy of the ``A'' band of molecular oxygen near 762 nm to provide a sensitive experimental test of the symmetrization postulate of quantum mechanics. We place an upper limit of $(0.8\ifmmode\pm\else\textpm\fi{}1.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ on the probability of finding two identical 16O nuclei in an antisymmetric state. This is one of the first sensitive tests of the symmetrization postulate for spin-0 particles.