The 23-year-old full-back suffered the injury in the 20th minute of Saturday's draw with Sunderland, and was immediately taken to hospital.
The Wales international has since had surgery and it has been confirmed that he has broken and dislocated his ankle in three places.
"The surgeons worked hard on Neil's injury last night and did a very good job," said Swansea's head physio Kate Rees on the club's official website.
"He is comfortable and the surgeons are happy with the operation.
"While he should make a full recovery, the extent of the injury means Neil is likely to miss the rest of the season."