It is a record signing for Victor Nilsson Lindelöf's club.
The Senegalese teenager became the youngest African World Cup goalscorer of all time this summer when he scored against France.
Mbaye made 31 appearances for PSG last season as the Paris club won both the Champions League and the domestic Ligue 1 title.
Aston Villa, who finished fourth in the league last season, have sold several of their stars during the summer and have been held to two consecutive defeats at the start of the season.
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