  The empirical value of the real part of the pion-deuteron scattering length
can be well understood in terms of the dominant isovector $\pi N$-double
scattering contribution. We calculate in chiral perturbation theory all
one-pion loop corrections to this double scattering term which in the case of
$\pi N$-scattering close the gap between the current-algebra prediction and the
empirical value of the isovector threshold T-matrix $T_{\pi N}^-$. In addition
to closing this gap there is in the $\pi d$-system a loop-induced off-shell
correction for the exchanged virtual pion. Its coordinate space representation
reveals that it is equivalent to $2\pi$-exchange in the deuteron. We evaluate
the chirally corrected double scattering term and the off-shell contribution
with various realistic deuteron wave functions. We find that the off-shell
correction contributes at most -8% and that the isovector double scattering
term explains at least 90% of the empirical value of the real part of the $\pi
d$-scattering length.
