  We compute the ${\cal O}(\alpha_s^3)$ virtual QCD corrections to the
$\gamma^*\to q\bar q g$ matrix element arising from the interference of the
two-loop with the tree-level amplitude and from the self-interference of the
one-loop amplitude. The calculation is performed by reducing all loop integrals
appearing in the two-loop amplitude to a small set of known master integrals.
Infrared and ultraviolet divergences are both regularized using conventional
dimensional regularization, and the ultraviolet renormalization is performed in
the $\bar{MS}$ scheme. The infrared pole structure of the matrix elements
agrees with the prediction made by Catani using an infrared factorization
formula. The analytic result for the finite terms of both matrix elements is
expressed in terms of one- and two-dimensional harmonic polylogarithms.
