But beyond surveys and tape-cuttings, the Koreas can not move much further along without the removal of US -led sanctions against the North, which isn't likely before it takes firmer steps toward relinquishing its nuclear weapons and missiles.
The two countries are considering three sites for the potential summit, Trump said on Saturday, but he did not offer up details about where those sites were.
Trump and Kim agreed to work toward the "complete denuclearization of the Korean Peninsula" during their first meeting on June 12 in Singapore.
While the United States allowed for the exemption, Washington is unlikely to support the further delivery of benefits to North Korea as far as plans for an inter-Korean rail connection are concerned.
During his meeting with Trump on Friday, Moon discussed the idea of Kim's visit to Seoul. The postponement stirred concerns that nuclear talks between the USA and North Korea had again bogged down, as Kim's regime seeks sanctions relief before taking key disarmament steps.
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The predominant modes of transmission in these countries were heterosexual transmission and transmission through injecting drug use.
The pair "reaffirmed their commitment to achieve the final, fully verified denuclearisation" of North Korea, Trump's spokeswoman Sarah Sanders said.
Already with Japanese Prime Minister Shinzo Abe, Trump spoke about the need to work together with the worldwide community to "keep up the pressure until North Korea implements its commitment to denuclearization". In recent months, however, the two sides have begun to remove landmines and destroy military bunkers at parts of the border as part of efforts to improve long-strained relations.
Trump, speaking aboard Air Force One on his return from Argentina, tells reporters that he and Kim are "getting along very well" and have a "good relationship".
They have also begun work to reconnect a train line and fix another rail link across the border. Moon and Kim specifically agreed to hold a tape-cutting ceremony by the end of this year to mark the beginning of an inter-Korean project connecting roads and railways between the two countries. The train is running on a track that leads to Kaesong, a North Korean city near the border with the South, to start a joint survey with Pyongyang on North Korean railway systems.
Appreciating Kim's upcoming visit to Seoul, Trump noted that the trip may create new momentum for efforts to denuclearize North Korea and establish peace on the Korean Peninsula, according to Yonhap News Agency.