The switch to Cardiff comes as Swansea City host Middlesbrough at the ground in the Premier League on the same day.
The Swans' game was moved to a Sunday by television broadcasters.
Paul Clement's side kick off at 13:30 GMT while Ospreys are due to face Stade later that day at 17:45 GMT.
There had been uncertainty over where Ospreys would play their quarter-final tie after they qualified as top seeds for the knock-out stages of European club rugby's second-tier competition.
Tournament organisers European Professional Club Rugby had confirmed the dates and times of all eight quarter-finals in the Champions Cup and Challenge Cup.
Ospreys were forced to look at contingency plans away from the Liberty Stadium with the possibility of Swansea's game against relegation rivals Middlesbrough being switched to Sunday to be televised live.
The Principality Stadium and the Cardiff City Stadium were options but it understood Ospreys reluctant to move the match against the French side to the closer venue of Parc y Scarlets in Llanelli.
"It's a positive move. It's our first home European quarter-final and a huge match. We looked at all the options on the table," Ospreys commercial manager Huw Warren told BBC Wales Sport.
"Clearly it's the best option both for on the field with Steve [Tandy, head coach] and the team but it's also a commercial opportunity for us in Cardiff for what is a massive game.
"Ultimately it's a massive opportunity for us as a brand to take the game there and hopefully our supporters understand and come and support us."
Ospreys have had to move their European Challenge Cup quarter-final against Stade Français from the Liberty Stadium to the Principality Stadium on Sunday, 2 April.