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The first leg of the tie, which will take place in the Netherlands, is scheduled for Thursday 10 March at 18:45 CET.
The second leg will take place exactly a week later at the Stadio Olimpico.
Kick-off on this occasion will be at 21:00 CET.
This will be the first time the two clubs have ever faced each other in European competition.
The Giallorossi are unbeaten in their last five games against Dutch opposition, having most recently defeated Ajax in the quarter-finals of last season's Europa League.
“I really like the draw, I think the whole team like it,” Vitesse coach Thomas Letsch said.
“Of course it is the toughest opponent we could get, but in the end it is exactly what we would have wanted."
Elsewhere, eye-catching ties include Premier League side Leicester City against French club Stade Rennais, while Swiss side Basel will go up against Olympique Marseille.
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