  An "ultrafine" splitting in positronium between the $L=1$ spin-singlet state
and the spin-average of the spin-triplet states is shown to arise only at order
$\alpha^6$. The QED prediction for $n=2$ states is
$\Delta_{2,P}=(683/172800)m\alpha^6=73.7(1.2)$ kHz. This represents the
smallest leading-order QED splitting known. Current experimental efforts could
observe this splitting, and its observation can constrain new ultralight
interactions, such as axions or $Z'$.
