  We have calculated the contribution to the positronium hyperfine splitting at
order $m \alpha^7$ of the light-by-light scattering process in the exchange of
two photons between the electron and positron. Our result is $\Delta E =
-0.235355(8) m \alpha^4 \left ( \frac{\alpha}{\pi} \right )^3 = -1.034 kHz$. As
a check of our approach we confirm earlier evaluations of the analogous
correction for a bound system (such as muonium) with unequal masses.
