That set in motion a set of highly choreographed actions in both Russia and the United States, all done with strict secrecy.
Two days ago, Russians moved Ms. Griner from the penal colony to Moscow, in preparation for a possible swap, according to a senior White House official. On Thursday morning, Russians put Ms. Griner on a plane and flew her to an airport in the United Arab Emirates, whose government had agreed to serve as the location for the swap.
In a video posted by the Russian state news agency RIA Novosti, Ms. Griner is shown boarding a plane in the snow with two duffel bags. Once aboard and in her seat, she is asked by an unidentified person, “What’s your mood?”
She replies, “Happy.”
Late on Wednesday afternoon, American officials put Mr. Bout on a plane headed to the U.A.E. Another video released by Russia’s state media shows the moment of the swap. Ms. Griner, wearing a red jacket and trousers, walks with three men in suits toward another man and Mr. Bout, who is clutching a brown envelope and holding a coat over one arm.
One man shakes Mr. Bout’s hand and embraces him, and then a second man also shakes his hand. Ms. Griner then veers off toward another group of men, including what appears to be an U.A.E. official clad in flowing robes, as they shake her hand.
Once American officials confirmed that Ms. Griner was ready to return to the United States, the president signed a conditional grant of clemency to Mr. Bout, who was captured in Bangkok in 2008, extradited to the United States in 2010 and sentenced in 2012 to 25 years in prison.
Around the time that Mr. Bout took off, the administration sent an official to Chappaquiddick, Mass., to let Elizabeth Whalen, his sister, know what was about to happen, a step they did not take when the government agreed to a similar swap in April for Trevor Reed, an ailing U.S. Marine veteran held for two years on what his family considered to be bogus charges of assault.
