Sotheby's refused to release any details about the buyer but confirmed to ATG that the new owner is now likely to loan Portrait of Catrina Hooghsaet to a UK institution.
The Art Fund, which had begun a campaign to acquire the work for the nation, released a statement saying it welcomed the news that the picture would not be "immediately leaving the country" but added that the opportunity to secure the portrait "may now never be achievable".
A £12.5m tax concession would have applied to the current sale, effectively enabling a UK body to match the purchase price by offering £22.5m.
The Art Fund statement says: "This case clearly highlights a pressing need for major improvements to the systems and procedures that are intended to protect our nation's treasures."
The Rembrandt, acquired by the 1st Baron Penrhyn in 1860 and hung at Penrhyn Castle, has been in Wales for 150 years. If the campaign had proved to be successful, plans were in place to donate it to Amgueddfa Cymru (National Museum Wales) and funds from charitable sources had apparently already been raised after the temporary export bar was announced on October 16.
Negotiations to sell the painting first took place in 2007 between the owners (the Douglas-Pennant family) and the Rijksmuseum. The museum had purchased Jan Steen's The Burgomaster of Delft for £8.1m from the family in 2004 but this time their £26m offer for the Rembrandt failed to meet the asking price.
Other works formerly in the Douglas-Pennant collection remain at Penrhyn Castle after being allocated to the National Trust in 2001 in lieu of inheritance tax. The family now intend to "explore the possibility of offering in lieu of tax other paintings in their ownership that are currently on loan to Penrhyn Castle", according to Sotheby's.
The announcement that Portrait of Catrina Hooghsaet will remain in the UK follows the news that two other Rembrandt portraits have been jointly acquired by the Louvre in Paris and the Rijksmuseum.
Previously owned by the Rothschild family, the 1634 paintings of Maerten Soolmans and his wife Oopjen Coppit, cost €160m (£135m) and will be shared between the two countries.
Catrina Hooghsaet was a wealthy Amsterdam woman who, at the time of the painting, was married but separated from her husband.
In Rembrandt's portrait she is accompanied not by her estranged husband but by her pet parrot which is visible in the background and features in her will.
She was a Mennonite (a member of an Anabaptist Christian group) and the portrait clearly reflects her strength of character and independence.
The picture dates from 1657, 12 years before Rembrandt died, at a time when he returned to portraits due to his financial difficulties. Whereas other late portraits were executed in the artist's 'rough manner', the precision of this work recalls a style that was present in his earlier portraits from the 1630s.
It recently featured at the Rembrandt: The Late Works exhibition at the National Gallery in London which then transferred to the Rijksmuseum in Amsterdam.