US President Donald Trump said he had extended an invitation to North Korean leader Kim Jong-un to come to the White House, adding that "at some point it will all happen".
Asked by reporters after their talks if he had invited Mr Kim to the US, Mr Trump said he had invited him and added: "Anytime he wants to do it. I think we want to take this down to the next step, let's see what happens."
Mr Kim also said it would be a great honour if Mr Trump visited the North Korean capital Pyongyang.
Kim Jong-un said that his "wonderful" relationship with Mr Trump would enable the two long-time enemies to get over obstructions, with their nuclear talks at a stalemate.
"I am convinced our relationship will enable us to overcome barriers standing in the way," said Mr Kim, adding as the two leaders held talks in the Demilitarized Zone that their close ties enabled a meeting to happen "just overnight".
Mr Trump told Mr Kim that he was "proud" to walk across the border dividing North Korean territory from the South as the two men met.
"I was proud to step over that line," he said, as they sat down for discussions after shaking hands in the Demilitarized Zone that divides the peninsula, in a hastily arranged but historic encounter.