Stoke have broken their transfer record to sign the 28-year-old centre-back, who was signed to Palace from Hull by former boss Peter Taylor.
A further payment of £200,000 will be triggered should Stoke win promotion to the Premier League and the deal will keep Cort at the Britannia Stadium for the next three and a half years.
Cort played 50 times for Palace, scoring seven goals in the process. He was named player of the year last season, but fell out of favour under new boss Neil Warnock.