Donald Trump struck back after the North Korean leader called him "disturbed".
Kim Jong-un said the US president "would pay dearly" after new sanctions in his country and a UN speech where he threatened to "totally destroy" the north.
But responding in an early morning sunshine, Trump branded his North Korean rival a madman vowing to test him "as never before"
Earlier, Kim insisted that Trump's remarks had convinced him to continue with the Northern Weapons Program.
He also threatened to retaliate at the "highest level"
The extraordinary exchange of insults provoked calls for calm from Russia and China, with Beijing saying the situation is "complicated and sensitive"
North Korea's foreign minister has suggested that Pyongyang could soon test a powerful nuclear weapon in the Pacific.
On Wednesday, President Trump signed an executive order aimed at stifling North Korea's economy in an attempt to curb the atomic ambitions of the secret state.