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The Giallorossi were drawn against the Dutch side during Friday's draw in Nyon.
The first leg of the tie will take place on Thursday 8 April, at the Johan Cruyff Arena. The second leg, at the Stadio Olimpico, will follow a week later.
The winner of the tie will face either Manchester United or Granada in the semi-finals.
The two sides have met just twice before in their history, in the Champions League back in the 2002-03 season - on that occasion Roma lost one game and drew the other.
With the draw completed all the way to the final in Gdansk, the Giallorossi now know they cannot face Arsenal, Slavia Prague, Dinamo Zagreb or Villarreal until that point.
"Ajax have a very attractive style of football which is so embedded in their history," Roma's General Manager, Tiago Pinto, said.
"This tie will undoubtedly be the most intriguing one in the quarter-finals, as both teams look to play an attractive and aggressive style of football."
Both legs will kick off at 21:00 CET.
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