The rate for the production of a pair of massive fermions in $e^+ e^-$
annihilation plus real or virtual radiation of a pair of massless
fermions is calculated analytically.
The contributions for real and virtual radiation are displayed separately.
The asymptotic behaviour close to threshold and for high energies
is given in a compact form. These approximations provide
arguments for the appropriate choice of the scale in
the ${\cal O}(\alpha)$ result, such that no large logarithms remain in the
final answer.