NLO QCD corrections to the production of off--shell WW pairs at $e^+ e^-$ colliders

QCD corrections to $e^+e^- \ra WW \ra q_1 \bar q_1 q_2 \bar q_2$ are computed and presented in a form which allows to impose realistic cuts on the structure of the observed events. QCD radiation substantially modifies the jet--jet invariant mass distributions from which the value of the $W$ mass will be extracted. When the range of allowed jet--jet masses is restricted, as it has been proposed in order to suppress non--$WW$ backgrounds, the total cross section is also deeply affected. If all events are forced to four jets, combining the two partons with smallest invariant mass, and the reconstructed masses are required to satisfy $ \left| M_{Ri} -M_W \right| \le 10$ GeV $i$=1,2 the lowest order cross section can be reduced by more than 40\%.