U.S. President Donald Trump and South Korean President Moon Jae In agreed Monday to closely coordinate over a second U.S.-North Korea summit that Trump said will take place "in the not too distant future."

In a meeting in New York, Trump and Moon pledged to maintain "vigorous enforcement" of sanctions on North Korea so as to ensure it "understands that denuclearization is the only path to economic prosperity and lasting peace on the Korean Peninsula," according to the White House.

Trump commended Moon for a "successful" summit with North Korean leader Kim Jong Un last week in Pyongyang, and said there remained much work to be done to accomplish their mutual goal of achieving the final, fully verified denuclearization of North Korea, it said.

Speaking to reporters after the meeting, Trump said, "I'm going to be meeting with Chairman Kim in the not too distant future. The location is being worked on, the time is being worked on, we'll be announcing it."

Trump suggested a second meeting with Kim -- chairman of the North's ruling party, the Workers' Party of Korea -- is likely to take place in a location other than Singapore where they held a historic summit in June.

Moon conveyed a private message from Kim to Trump, and the U.S. and South Korean leaders exchanged views on the possible timing and location of a Trump-Kim summit, according to an official of South Korea's presidential office.

Trump and Moon also discussed Kim's desire to secure a declaration on ending the 1950-1953 Korean War -- which concluded in an armistice, not a peace treaty -- as a way of building trust and guaranteeing the North's security.

The U.S. and South Korean leaders met on the sidelines of the U.N. General Assembly after Kim told Moon last week that he was willing to have a second meeting with Trump soon to push ahead with denuclearization negotiations.

Moon told Trump that Kim reaffirmed his commitment to denuclearization, and that Kim's "decision to relinquish its nuclear program has been officialized to a degree that not even those within North Korea can reverse."

"Chairman Kim also repeatedly conveyed his unwavering trust and expectations for you, while expressing his hope to meet you soon to swiftly conclude the denuclearization process with you, because you are, indeed, the only person who can solve this problem," Moon told Trump at the meeting, part of which was open to the media.

Earlier Monday, U.S. Secretary of State Mike Pompeo expressed eagerness to travel to North Korea "before the end of the year" to make "final preparations" for a second summit.

During two-day talks through last Wednesday with Moon in Pyongyang, Kim pledged to permanently dismantle North Korea's key missile test site in Tongchang-ri under the watch of international experts, and to do the same to its main nuclear complex in Yongbyon if the United States takes reciprocal measures.

Moon's trip to the North was intended to break the impasse over U.S.-North Korea negotiations on Pyongyang's abandonment of its nuclear and missile programs.

The talks began after Kim committed to "complete" denuclearization of the Korean Peninsula in the first-ever U.S.-North Korea summit.

The United States has criticized North Korea for failing to take credible measures to give up its nuclear weapons program. Pyongyang has insisted it first wants a declaration on ending the Korean War as a way of building trust and guaranteeing its security.

The 1950-1953 conflict -- involving the U.S.-led United Nations Command on one side and North Korean and Chinese forces on the other -- ended in an armistice, not a peace treaty.

After the latest inter-Korean summit, Pompeo expressed readiness to "immediately" engage in negotiations with North Korea in a bid to achieve the goal of denuclearizing Pyongyang by the end of Trump's first term in January 2021.

As part of efforts to jump-start stalled denuclearization talks, Pompeo said he invited North Korean Foreign Minister Ri Yong Ho to meet while in New York for the U.N. General Assembly.

Aside from the discussions on North Korea, Trump and Moon on Monday signed a revised U.S.-South Korea free trade agreement, which Trump said will help reduce the U.S. trade deficit with Seoul and expand opportunities to export American products to South Korea.

"These outcomes give the finest American-made automobiles, innovative medicines and agricultural crops much better access to Korean markets," Trump said.

Under the revised FTA, South Korea will double the annual number of American automobiles that can enter its market using U.S. safety standards without further modifications from 25,000 to 50,000 per manufacturer a year, according to the White House.