OSAN AIR FORCE BASE, South Korea — President Trump became the first sitting U.S. president to set foot on North Korean soil Sunday in a dramatic border meeting with Kim Jong-un that reopened denuclearization talks, capping a momentous Asia trip in which the president also revived trade negotiations with China.

Mr. Trump stepped across the demarcation line in the heavily guarded Demilitarized Zone that separates North and South Korea, shook hands with Mr. Kim and said it was a “great honor” to cross the border. As he did so, both leaders agreed to resume stalled talks aimed at compelling Pyongyang to abandon its nuclear weapons and missile systems.

“I think it’s historic. It’s a great day for the world,” Mr. Trump said.

Mr. Kim, dressed in his typical Mao suit, greeted Mr. Trump warmly. He told the president that he “never expected to meet you at this place.”

“Meeting at such [a] place shows that we are willing to put an end to the unfortunate past and also open a new future,” he said through an interpreter.

It was their first face-to-face meeting since a second summit fell apart in Hanoi, Vietnam, in February. The president walked away from that meeting because the North Koreans demanded sweeping sanctions relief in exchange for only a limited commitment to destroy part of their nuclear arsenal at the Yongbyon nuclear complex.

Mr. Trump hinted at a softening of that approach.

“The sanctions remain, but at some point during the negotiations, things can happen,” the president said. “We’re not looking for speed. We’re looking to get it right. We’re on a very good path.”

Based on the impromptu meeting at the DMZ, Secretary of State Mike Pompeo said Stephen E. Biegun, the U.S. special representative for North Korea, will conduct talks with his North Korean counterparts by mid-July.

He said the parties are not starting at square one despite the narrative that both sides are stuck.

“Even in Hanoi, we made progress. So we think we do have a jumping-off point for these discussions, which will put us in a place where we can truly evaluate if there’s a clear path forward,” Mr. Pompeo told reporters on his plane. “Having listened to Chairman Kim today, I think there is.”

He added, “We thought there might be this opportunity and the president, by getting together with Chairman Kim today, broke through and was able to get us the opportunity to get back to the negotiating table.”

Gen. Robert B. Abrams, commander of U.S. Forces Korea, said “the mission remains the same” while diplomacy unfolds.

“We have to maintain our edge,” he told soldiers at Osan Air Base south of Seoul. “We’ve got to be ready to fight tonight.”

In the U.S., Democratic presidential candidates belittled Mr. Trump’s effort. A spokesman for former Vice President Joseph R. Biden said the president “fawned” over Mr. Kim. Mr. Biden also criticized Mr. Trump for “coddling dictators” on his five-day trip.

Rep. Tim Ryan of Ohio called the president’s trip an “appeasement tour.” He compared Mr. Trump’s meeting with Mr. Kim to British Prime Minister Neville Chamberlain “going to talk to Hitler” and making concessions before the start of World War II.

Sen. Bernard Sanders of Vermont had no objection to the meeting.

“I have no problem with him sitting down with Kim Jong-un in North Korea or anyplace else, but I don’t want it simply to be a photo opportunity,” Mr. Sanders told ABC News.

Mr. Trump risks legitimizing a ruthless strongman who, many say, will never give up his nuclear ambitions.

Mr. Kim recently tested short-range missiles, and a failed summit or provocation that could harm the U.S. would damage Mr. Trump on the cusp of his reelection campaign.

The unexpected round of diplomacy came after Mr. Trump announced on social media, while attending the Group of 20 summit in Japan, that he hoped Mr. Kim would meet him at the DMZ. The North Korean leader said he was “surprised” by Mr. Trump’s public overture.

The two leaders met at the truce village of Panmunjom, where the armistice was signed to pause the Korean War in 1953.

After they shook hands across the boundary line, Mr. Trump walked about 20 steps into the North with Mr. Kim, igniting a media frenzy as U.S. journalists shouted at North Korean photographers to stop blocking their view of the action.

The two leaders then returned to the demarcation line, and the president asked Mr. Kim a question.

“Would you like to come over?” Mr. Trump asked, gesturing to the South.

Mr. Kim agreed and walked into the Freedom House building with Mr. Trump on the South Korean side.

They spoke for a total of 66 minutes, signaling they had more on their minds than a photo-op handshake. They emerged with the commitment to restart nuclear talks.

Mr. Trump said North Koreans on site were in tears as he crossed the border.

It was the third encounter between the two men, after a pair of summits that forged a personal bond but failed to produce a pact that deals with Mr. Kim’s nuclear and missile systems.

“We’re all in this together. We want to get this thing solved,” he told cheering U.S. soldiers at Osan.

The president even said he would like to invite Mr. Kim to the White House.

Mr. Trump exchanged warm letters with Mr. Kim before this visit to Asia, giving South Korean President Moon Jae-in hope that he could seize on that momentum to advance the Korean peace process.

Mr. Moon, who met with Mr. Trump in Seoul, played the role of mediator over the weekend. He is urging Mr. Trump to be flexible because he has tied his own legacy to making progress on Korean peace.

“You really are the peacemaker of the Korean Peninsula,” Mr. Moon told Mr. Trump.

Mr. Moon also joined Mr. Trump at the DMZ, which is 160 miles long and 2.5 miles wide.

Previous U.S. presidents, including Barack Obama, have visited the DMZ to reaffirm the U.S.-South Korean alliance, though Mr. Trump is the first to use the backdrop as an embrace of the North. President Clinton once dubbed it the “scariest place on earth.”

Some folks in South Korea weren’t happy.

Cho Won-jin, the leader of the far-right Our Republican Party, said regime change is the only way to solve the North Korea problem. He said Mr. Trump was making a mistake by agreeing to a trilateral visit at the DMZ.

He said Mr. Moon’s progressive government is “socialist” and an unreliable partner with America.

“We oppose President Trump meeting Kim Jong-un at the DMZ,” he told The Washington Times late Saturday. “It is very dangerous for President Trump to take [Mr. Moon’s] side.”

By late Sunday, however, Mr. Trump was on a victory lap, saying Mr. Moon called him on the way to Osan.

“He’s so happy,” Mr. Trump said. “He’s so thrilled.”

It was Mr. Trump’s second rescue of stalled major negotiations on his trip. At the G-20 summit in Osaka, Japan, on Saturday, he agreed with Chinese President Xi Jinping to resume trade talks that collapsed in acrimony in May when the U.S. accused Beijing of reneging on commitments.

Mr. Trump said his meeting with Mr. Xi went “far better than expected” and that he agreed not to impose tariffs on some $300 billion worth of Chinese imports to the U.S. He also agreed to allow U.S. high-tech companies to resume selling components to Chinese telecommunications giant Huawei after blacklisting the firm over national security concerns.

In return, Mr. Xi agreed that China will purchase more U.S. agricultural products, Mr. Trump said.

“We’re right back on track,” Trump told reporters. “We’re holding back on tariffs, and they’re going to buy farm products.”

• Dave Boyer contributed to this report from Washington, D.C.

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