Trump responded with a tweet calling Kim Jong-un "a madman" who "will be tested like never before!"
Experts say this is the first time a North Korean leader has made a direct address to an worldwide audience - and it merits serious and thorough consideration.
But the North Korean leader said remarks by the "deranged" US President convinced him he is right to develop weapons for North Korea.
He called Mr Trump "mentally deranged" and said the US President's comments were "the most ferocious declaration of war in history".
The threatening message came hours after the rogue nation's foreign minister said that North Korea could test a powerful nuclear weapon over the Pacific Ocean in response to Trump's threats of military action.
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Some advisers now worry that the escalating war of words has pushed the impasse with North Korea into a new and unsafe phase that threatens to derail the monthslong effort to squeeze Pyongyang's economy through sanctions to force Kim to the negotiating table.
On the fringes of the General Assembly, North Korean Foreign Minister Ri Yong-ho told reporters Pyongyang might now consider detonating a hydrogen bomb outside its territory.
Trump also praised China on Thursday for what he said was instructions to its banks to cut off business with North Korea.
In a rare statement, Kim said that Trump would "pay dearly" for his threat, a state media report said. In their statements, Trump and Kim combined insults with threats. Kim had delivered a tongue-lashing of his own - vowing to "surely and definitely tame the mentally deranged U.S. dotard with fire", in an address read out on state television by a star news anchor before a still image of Kim at his desk.
Kim confirmed his nuclear program was on the "correct path" this week. "Because we really have no choice, we really have no choice", he said.
It's the latest escalation in a suddenly personal war of words between the two leaders that's stretched back weeks, unfolding as North Korea has showcased an enhanced nuclear missile capability. The council has told member countries to ban most activity overseas by North Korea's banks in response to its nuclear and missile tests.