Celtic have described it as the "biggest transfer" in their history as Odsonne Edouard agreed a four-year contract from Paris St-Germain.
BBC Scotland believes Paris St-Germain were hoping to receive a fee in excess of £8m for the French forward, 20.
Edouard scored 11 goals in 29 appearances on loan from PSG last term.
Announcing the transfer, manager Brendan Rodgers said: "Odsonne is one of the best young players I have worked with and one of the best in Europe."
Edouard came through the youth ranks with PSG and, while on loan to Toulouse in season 2016-17, was sentenced to a suspended four months' prison sentence after an alleged firearms incident also involving then team-mate Mathieu Cafaro.
But, having joined Celtic the following season, he helped the Glasgow club secure a second successive domestic Treble.
"Our fans saw exactly what he is capable of last season and we look forward to working with him again as we look to build on the tremendous success we achieved last year," Rodgers told Celtic's website.
"I am sure he will be a big player again for us this season."
Edouard said the scenes that followed their Treble win persuaded him to stay in Glasgow.
"I was desperate to make my move permanent," Edouard said.
"To see our fans so happy is the reason you play football and now we want to do everything we can to bring our great supporters even more success."
Celtic's previous record transfer fee was £6m for Chris Sutton and John Hartson.
Ligue 1 champions PSG need to sell £52.9m-worth of players by the end of June to comply with financial fair play (FFP) rules.
Uefa introduced 'break-even' rules in 2013, requiring clubs to balance their spending with their revenue.
Edouard becomes Celtic's second summer capture after goalkeeper Scott Bain signed a permanent deal after also spending time on loan in Glasgow.
Rodgers' side will begin season 2018-19 with Champions League qualifiers next month and open their Scottish Premiership title defence at home to promoted Livingston on 4 August.
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