  An effective theory of the excited states of positronium is derived and some
of its consequences are explored. At large physical separation, the binding of
the electron and positron is assumed to be described completely by QED, whereas
all short-ranged phenomena, including those within and beyond QED, can be
accounted for with energy-dependent quantum defects. This theory has at least
two practical applications. First, it provides an accurate and economical, yet
largely QED-independent, means to fit the positronium spectrum in order to
predict and compare the outcome of experiments. Second, matching the spectrum
in this effective theory to that predicted by QED reveals nontrivial
relationships that exist \emph{within} bound-state QED; some higher order
contributions to the spectrum may be obtained from lower order contributions.
These relations are verified up to order $m\alpha^{6}$, and predictions are
made for the order $m\alpha^{7}$ and $m\alpha^{8}$ corrections. This theory and
its extensions to other hydrogenic systems may provide a useful complement to
bound-state QED.
