  We compute the two-loop QCD helicity amplitudes for the process e^+e^- --> q
bar{q} g. The amplitudes are extracted in a scheme-independent manner from the
coefficients appearing in the general tensorial structure for this process. The
tensor coefficients are derived from the Feynman graph amplitudes by means of
projectors, within the conventional dimensional regularization scheme. The
actual calculation of the loop integrals is then performed by reducing all of
them to a small set of known master integrals. The infrared pole structure of
the renormalized helicity amplitudes agrees with the prediction made by Catani
using an infrared factorization formula. We use this formula to structure our
results for the finite part into terms arising from the expansion of the pole
coefficients and a genuine finite remainder, which is independent of the scheme
used to define the helicity amplitudes. The analytic result for the finite
parts of the amplitudes is expressed in terms of one- and two-dimensional
harmonic polylogarithms.
