  The influence of the hadronic \pi\pi interaction on the wave functions of
pionium (the Coulomb bound \pi^+\pi^- system) at distances less than 10 fm is
calculated for s-states with principal quantum numbers n<5. Hadronic \pi\pi
potentials are used that reproduce the scattering phase shifts of two-loop
chiral perturbation theory. The pionium wave functions \psi_n (r) are obtained
by integrating the coupled Schr\"odinger equations for the
(\pi^+\pi^-,\pi^0\pi^0) system. We find that, for r<10 fm, n^{3/2} \psi_n (r)
is practically independent of n. From this we conclude that the production
rates of the s-states of pionium are proportional to n^{-3}, a result needed
for the interpretation of the DIRAC experiment currently running at CERN.
