  The contribution of positronium to the electron $g$-2 ($a_e$) has been
computed in Ref. [1] and found to be of the same order of $\alpha$ as that of
five-loop perturbative QED. We confirm this result and correct a few errors in
its first derivation. As recently calculated in Ref. [2], a continuum
nonperturbative contribution to $a_e$ cancels one-half of the positronium one.
We show by explicit calculation that the remaining half is already included in
the five-loop perturbative result. We also show that it arises from the class
I(i) of five-loop diagrams containing only one closed electron loop.
