The 26-year-old left back, who has played in three of the club's four Premier League games this season, was involved in an accident on Sunday and needed to be cut free from his car.
"Thanks to the fantastic response of the emergency services who cut him from the car and the amazing surgeons, doctors and staff at the Royal London Hospital, he is expected to make a full recovery and should be discharged on Friday," chairman Steve Parish told Palace's website (www.cpfc.co.uk) on Tuesday.
"We would like to make special mention of the London Air Ambulance service who flew Pape directly from the scene to the hospital ensuring that he rapidly had access to the very best medical care."
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(Reporting by Martyn Herman; Editing by Tony Jimenez)
