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Jason Levien and Steve Kaplan, experienced in running US sports teams, lead a consortium which has bought a controlling stake of 68% in the Swans.
The new owners, who received Premier League ratification on 1 July, finalised the deal in Swansea following a meeting with chairman Huw Jenkins.
Swansea Supporters Trust retain a 21% shareholding in the club.
The new Swansea majority shareholders broke the news in an open letter to supporters.
They had been due to hold a press conference on Thursday afternoon, but it was delayed due to a dispute between minority shareholders.
Although retaining its stake, the Supporters Trust was 'disappointed' about not being included in any talks.
The Trust feels the new owners have yet to give any indication of "investment into the football club, if any, is proposed", adding that it wants to discuss long-term plans for the club and the negotiation of a new shareholders' agreement.
After receiving Premier League approval earlier this month, Levien and Kaplan said they had "minor loose ends to tie up", which appear to have been resolved following Wednesday's meeting.
Jenkins will stay in his role under the terms of the agreement, as will vice-chairman Leigh Dineen.
BBC Wales Sport revealed in April which of Swansea's directors would be selling all - or part - of their shares.
Levien is the managing general partner of Major League Soccer side DC United, while Kaplan is principal of Oaktree Capital investment fund and vice-chairman of NBA franchise Memphis Grizzlies.
They are understood to be keen to buy Swansea's home ground, Liberty Stadium, which is owned by the local council, with a view to increasing its capacity.
Levien and Kaplan had initially been negotiating a deal which would have seen them acquire more than 75% of Swansea's shares, effectively giving the American consortium complete control, including the power to issue more shares.
However, the modified acquisition of 60% will see the trust retain its 21.1% stake and ensure continuity at board level with the retention of Jenkins and Dineen.
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An American consortium has completed the purchase of a controlling stake in Premier League club Swansea City.