"We can be successful in the denuclearisation of the Korean Peninsula, that could be a great thing for North Korea, it would be a great thing for South Korea, it would be great for Japan, it would be great for the world, it would be great for the United States, it would be great for China".
President Trump, meanwhile, said nothing about the surprise inter-Korean summit for hours.
"It's moving along very nicely", Trump said at the White House during a meeting with a U.S. prisoner freed by Venezuela.
With the Trump-Kim summit back on the cards, Moon said that he is looking forward to a three-way meeting between him, Trump and Kim to formally put an end to the Korean War.
Trump added: "We'll see what happens".
"While Washington and Pyongyang have expressed their hopes for a summit through published statements, Moon has to step up as the mediator because the surest way to set the meeting in stone would be an official confirmation of intent between heads of states", Hong said.
According to Reuters, US President Donald Trump said on Saturday that he was still looking at June 12 summit with North Korean leader Kim Jong-un.
During his two hour meeting with Kim, Moon said he urged both Washington and Pyongyang "to remove misunderstandings through direct communication and to have sufficient dialogue in advance through working-level negotiations on the agendas to be agreed upon at the summit".
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In a letter to Mr Kim on Thursday (local time), Mr Trump had said he was cancelling the planned Singapore summit, citing North Korea's "open hostility".
The summit took place from 3 p.m.to 5 p.m.at Panmungak, a North Korean building inside Panmunjeom, Yoon said.
It remains unclear whether Kim will ever agree to fully abandon his nuclear arsenal in return, despite Moon's insistence that Kim can be persuaded to abandon his nuclear facilities, materials and bombs in a verifiable and irreversible way in exchange for credible security and economic guarantees.
The two met secretly at the North's side of the border at the truce village of Panmunjom on Saturday afternoon, then later made the meeting public.
Asked if there is a plan to hold high-level inter-Korean talks to discuss follow-up measures to the April 27 summit agreement, the spokesman said that nothing has been determined.
"I urge North Korea and the U.S. to confirm each other's will by sharing their problems with each other and communicating directly".
"Kim stressed again that he had a firm determination toward complete denucleariaation", the South Korean president added.