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The scrum-half scored a try in the leaders' 40-24 win at the Principality Stadium on Sunday (AEDT), but also sustained an injury.
Hardy has returned to the Scarlets to continue his rehabilitation, with a weekend off to come before a trip to face bottom side Italy at Stadio Olimpico a week on Sunday.
Wales attack coach Stephen Jones said: "It's unfortunate [for Hardy] after having a wonderful performance against England.
"We've got three scrum-halves left who are good to go."
Jones also confirmed fly-half Biggar is carrying an injury, but is "very optimistic" he can face the Azzurri.
Aaron Wainwright and Owen Watkin have been released to play for the Dragons and the Ospreys respectively this weekend. The duo will return to link up with the national team after featuring for their regions.
Wales wrapped up the Triple Crown with a defeat of the defending champion and lead France, which has a game in hand against Scotland, by five points.
France and Wales are set to do battle at Stade de France on 21 March.