  The pseudoscalar pole contribution is the dominant source of the $ {\cal
O}(\alpha^3) $ hadronic light-by-light scattering effect in muon $g - 2$.
  We have examined this contribution taking account of the off-shell structure
of the pseudoscalar-photon-photon anomaly vertex deduced from available
experimental data.
  Our work leads to an improved estimate, $ -79.2 (15.4) \times 10^{-11}$, for
the total hadronic light-by-light scattering contribution to the muon $g - 2$.
