Mr Moon's office said that he had conveyed "Kim Jong-un's desire for a papal visit" and that a formal invitation from North Korea would follow.
South Korean President Moon Jae-in and his wife Kim Jung-sook, right, meet Pope Francis during a private audience at the Vatican. Pope Francis said if the North sends an official invitation, he will "certainly" respond to it.
On 17 October, the first talks between US, South Korean, and North Korean military representatives were held following summits between their leaders in the first half of 2018.
"President Moon explained our government's efforts to denuclearize the Korean Peninsula and establish peace, and President Mattarella highly evaluated President Moon's leadership in establishing peace on the Korean Peninsula", it said in a press release.
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South Korean President Moon Jae-in delivered remarks at a Wednesday evening "Mass for Peace" in St. Peter's Basilica.
North Korea's constitution guarantees freedom of religion as long as it does not undermine the state. Moon's office quoted the pope as telling Moon: "Do not stop, move forward".
US officials are concerned that the no-fly zone proposal, which is part of an agreement between North and South Korea reached during a summit last month in Pyongyang, could hinder defense readiness, according to Reuters, which cited two unidentified sources.
It spoke only of "the promotion of dialogue and reconciliation between Koreans" and "the common commitment to fostering all useful initiatives to overcome the tensions that still exist in the Korean Peninsula, in order to usher in a new season of peace and development".
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But beyond a handful of state-controlled places of worship - including a Catholic church in the capital of Pyongyang - no open religious activity is allowed and the authorities have repeatedly jailed foreign missionaries.
The pope said the verbal invitation relayed by President Moon should be sufficient but that an official invitation would also be nice, according to Yoon.
Following an unusually provocative run of weapons tests past year, Kim has been on a diplomatic offensive, which included three summits with Moon and the one with Trump.
Moon, who is Catholic, wrote that when he traveled to Pyongyang in September for the summit, he was joined by a Korean Catholic bishop to try to improve relations between the church in North and South.
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