"Kim Jong Un of North Korea, who is obviously a madman who doesn't mind starving or killing his people, will be tested like never before!" Trump tweeted.
Earlier today Kim blasted Trump as a "mentally deranged US dotard" who will "pay dearly" for threatening to destroy North Korea.
Trump "insulted me and my country in front of the eyes of the world and made the most ferocious declaration of a war in history", Kim said, according to the official Korean Central News Agency.
Trump branded Kim "Rocket Man" and threatened to "totally destroy North Korea" in his UN speech. Washington followed yesterday by announcing tougher sanctions on North Korea.
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Trump had earlier threatened Pyongyang with "fire and fury."
Trump has traded tough rhetoric with Pyongyang as it pushed an increasingly brazen weapons programme in recent months, with missile launches and its sixth and largest nuclear test.
On the fringes of the UN General Assembly in New York, North Korean Foreign Minister Ri Yong-ho told reporters Pyongyang might now consider detonating a hydrogen bomb outside its territory.