North Korea said on state television on Sunday afternoon that it successfully carried out a test of a hydrogen bomb meant to be carried by an intercontinental ballistic missile.
China's foreign ministry on Sunday also urged North Korea to stop its "wrong" actions, adding it urged the isolated nation to respect the U.N.
He then made a vague war threat by writing, "South Korea is finding, as I have told them, that their talk of appeasement with North Korea will not work, they only understand one thing!"
Just hours before the nuclear test, Trump talked by phone with Japanese Prime Minister Shinzo Abe about the "escalating" nuclear crisis.
French President Emmanuel Macron urged the United Nations Security Council to react quickly and decisively.
Moon said all available diplomatic actions, including new UN Security Council resolution, should be considered to completely isolate North Korea in response to the its sixth nuclear device test. Later on Sunday, Russian President Vladimir Putin was set to meet his Chinese counterpart Xi Jinping in China.
North Korea's announcement followed reports of as many as two tremors in the rogue state at around noon local time, which officials in South Korea and Japan had said appeared to be the country's sixth nuclear test.
The move to enhance South Korea's missile defence systems comes shortly after a missile was sacked by North Korea into the sea surrounding Japan.
The organisation in charge of the global nuclear test ban treaty - which can not come into legal force until ratified by the United States, China, India, Iran, Israel, Pakistan and Egypt as well as North Korea - said things had to change.