  The electrical {\em dc}-conductivity of disordered, non-interacting electrons
is calculated in the asymptotic limit of high lattice dimensions $d\to \infty$.
To go beyond the lowest-order contribution in the expansion parameter $1/d$ of
the single bubble diagram, vertex corrections are calculated from an asymptotic
expression for the two-particle vertex. A mean-field approximation for the
dc-conductivity containing the leading high-dimensional vertex corrections is
proposed which is free of spurious non-analyticities, i.e. the conductivity is
non-negative and shows no unphysical behavior in $d\geq 3$.
