  We calculate the high energy behavior of quark-antiquark exchange in
$\gamma^* \gamma^*$ elastic scattering by summing, to all orders, the leading
double logarithmic contributions of the QCD ladder diagrams. Motivation comes
from the LEP data for $\sigma_{tot}^{\gamma^* \gamma^*}$ which indicate the
need for secondary reggeon exchange. We show that, for large photon
virtualities, this exchange is calculable in pQCD. This applies, in particular,
to parts of the LEP kinematic region.
