The deal represents a profit of more than £500m for Advent, which bought Priory for £925m five years ago.
The move is part of a broader push into the UK by Tennessee-based Acadia, which operates 54 facilities in the country after buying Priory's rival, Partnerships in Care, for £394m in June 2014.
Joey Jacobs, Acadia's chief executive, said a long-term increase in demand for behavioural health services would provide opportunities for growth and further acquisitions in the UK.
Falling NHS funding has helped the UK's private behavioural healthcare market to grow 9.2 per cent a year since 2004.
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Acadia declined to comment on its UK market share, but the combined revenues of Priory and Partnerships in Care would give the company more than 50 per cent of the non-government market, according to estimates provided by the company in 2014.
Priory Group had revenues of £521m in 2014, the last year for which figures are available, and Acadia expected revenue to have grown to about £587m last year. Shares in Acadia rose 3 per cent during morning trading in New York.