  A quantum electrodynamics (QED) correction surface for the simplest
polyatomic and polyelectronic system H$_3^+$ is computed using an approximate
procedure. This surface is used to calculate the shifts to vibration-rotation
energy levels due to QED; such shifts have a magnitude of up to 0.25 cm$^{-1}$
for vibrational levels up to 15~000 cm$^{-1}$ and are expected to have an
accuracy of about 0.02 cm$^{-1}$. Combining the new H$_3^+$ QED correction
surface with existing highly accurate Born-Oppenheimer (BO), relativistic and
adiabatic components suggests that deviations of the resulting {\it ab initio}
energy levels from observed ones are largely due to non-adiabatic effects.
