  We compute the three-loop corrections to the helicity amplitudes for
$q\bar{q}\to Q\bar{Q}$ scattering in massless QCD. In the Lorentz decomposition
of the scattering amplitude we avoid evanescent Lorentz structures and map the
corresponding form factors directly to the physical helicity amplitudes. We
reduce the amplitudes to master integrals and express them in terms of harmonic
polylogarithms. The renormalised amplitudes exhibit infrared divergences of
dipole and quadrupole type, as predicted by previous work on the infrared
structure of multileg scattering amplitudes. We derive the finite remainders
and present explicit results for all relevant partonic channels, both for equal
and different quark flavours.
