Palace paid a reported £2million for Hunt on transfer deadline day, but the 22-year-old right-back faces up to four months on the sidelines after sustaining the injury only seven days after joining the London club.
Hunt wrote on Twitter: "Absolutely devastated to have broken my ankle in training (on Monday).
"Got to take the positives, clean break and no op needed."
Hunt, a product of the Huddersfield academy, added: "Massive thanks to everyone at CPFC. Chairman, manager, coaching staff, physios and the lads have been unreal."