The White House said Wednesday that President Trump isn’t wavering from his focus of forcing North Korea to give up its nuclear weapons after Pyongyang made a goodwill gesture of releasing three captive Americans, who were flying home for reunions with their families.

Even as Mr. Trump made plans to welcome home the three men at Andrews Air Force Base before dawn on Thursday, the White House emphasized that the positive development was only a precursor to the tough negotiations ahead with North Korean leader Kim Jong-un.

Vice President Mike Pence said that although the administration is “encouraged that North Korea freed these innocent hostages, we will not let off the pressure until we achieve full denuclearization.”

The president was triumphant about North Korea’s decision to release the American captives, whose freedom was secured by Secretary of State Mike Pompeo on his second trip in six weeks to the isolated communist nation.

“I appreciate Kim Jong-un doing this and allowing them to go,” Mr. Trump told reporters. “We’re honored by the fact that the three gentlemen are coming home. It’ll be 2 o’clock in the morning. It’ll be quite a scene. It’s very exciting because it represents something very important to this country.”

Mr. Pompeo said the three men “were all thrilled when we knew we were out of North Korean airspace. [They] were happy to be with us on this plane, to be sure.”

The longest-held prisoner was Kim Dong-chul, a 64-year-old formerly of Fairfax, Virginia, who was arrested in October 2015. He was sentenced in 2016 to 10 years in prison on charges of espionage and subversion.

Tony Kim, 59, had been detained at Pyongyang’s airport as he prepared to depart in April 2017. He was arrested for “committing criminal acts of hostility aimed to overturn” North Korea.

Kim Hak-song, an agricultural consultant, also was detained in the spring of 2017.

“I welcomed them back,” Mr. Pompeo told reporters traveling with him. “I’m thrilled that we have them back. President Trump set conditions for this to happen, and I am thrilled with that.”

Doctors were with the men on their flight out of North Korea. They transferred in Japan to another U.S. government plane with “even more robust medical capabilities … in the event they need it,” Mr. Pompeo said.

The men said in a joint statement: “We would like to express our deep appreciation to the United States government, President Trump, Secretary Pompeo, and the people of the United States for bringing us home.”

The secretary of state said his 13 hours in North Korea included “good, lengthy conversations” with the Communist dictator about the upcoming summit with Mr. Trump. He said there were no glitches during his visit, which took place at virtually the same time that Mr. Trump was ripping up the U.S. nuclear agreement with Iran.

Asked if Mr. Pompeo’s trip was meant to reassure North Korea about negotiating a denuclearization pact with Washington, White House press secretary Sarah Huckabee Sanders said the purpose of the meeting “was to make sure that we continued the progress that we have had on setting the date and the location” for the summit.

Mr. Trump told reporters that U.S. and North Korean negotiators have agreed on a date and location for the summit, which he said would be revealed within the next three days. He said the talks will not be held at the Demilitarized Zone separating North and South Korea, as he had proposed. Another possible site is Singapore, a neutral city-state in Southeast Asia.

“I think we have a really good chance to make a great deal for the world,” Mr. Trump said. But he conceded that a deal could be scuttled, depending on North Korea’s approach.

House Minority Whip Steny H. Hoyer, Maryland Democrat, commended Mr. Pompeo but not Mr. Trump for the diplomatic success.

“Releasing detainees is the easy part; the difficult part will be reaching an agreement with the North Koreans that establishes a strong verification regime that can ensure genuine denuclearization,” Mr. Hoyer said. “Congress has yet to see a concrete strategy from the president as to how he intends to avoid giving Mr. Kim a propaganda victory without securing the kind of verifiable commitment on denuclearization necessary for any summit to be a success.”

Human rights groups cheered the release of the Americans but said it shouldn’t obscure North Korea’s abysmal record.

“We remain concerned over the fate of up to 120,000 North Koreans who remain confined in political prison camps,” said Francisco Bencosme, an official with Amnesty International USA. “No one should languish behind bars without being given a fair trial. President Trump and his administration must urge North Korea to respect freedom of expression and other human rights at every opportunity, including the possible upcoming summit.”

The country’s top diplomat said he is “confident we set conditions for a successful meeting between the two leaders.” Mr. Pompeo said the date and location of the summit will be disclosed “in the next handful of days.”

The return of the captives eliminates a stumbling block to that summit, especially since it appears that the men who had been held in labor camps are in relatively good health. The White House said all three men “were all able to walk on the plane without assistance.”

The most recent U.S. citizen to be released from North Korean custody was Otto Warmbier, a student from Ohio who had been detained in 2016 for allegedly removing a propaganda poster from a wall.

Warmbier died in June 2017, just days after arriving home in a coma. A coroner determined that the 22-year-old died from lack of oxygen to the brain but couldn’t determine how he suffered his injuries.

Sen. Rob Portman, Ohio Republican, said although it’s a shame that the developments came too late to save Warmbier, Mr. Trump deserves all the credit for North Korea’s changed attitude.

“This a testament to the Trump administration’s diplomatic approach, which has balanced tough sanctions and united international pressure with openness to a dialogue with the North Korean regime,” Mr. Portman said. “This strategy helped create the space for today’s release.”

• Guy Taylor contributed to this article.

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