The Hwasong-15 missile is believed to have flown 1,000 km from North Korea before splashing down in the Sea of Japan, shattering a two-month calm in Northeast Asia.
The provocative action also came days after Trump announced fresh sanctions on Pyongyang and returned it to a US list of state sponsors of terror.
"I will only tell you that we will take care of it, Trump told reporters at the White House, alongside the Republican leaders in Congress.
Trump, who earlier this year warned he would rain "fire and fury" on the isolated state if it continued to threaten the US or its allies, was relatively muted this time in responding to Kim's latest provocation.
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North Korea's missile launch today was the first since the country fired an intermediate range missile in September.
The ICBM also reportedly went approximately 4,000 km up into the atmosphere, according to South Korean and Japanese authorities. US Defence Secretary James Mattis said the latest missile did fly higher than previous ones.
"It went higher, frankly, than any previous shot they've taken," Mattis told reporters.
"The US Department of Defense detected and tracked a single North Korea missile launch today at about 1:17 p.m. EDT," Pentagon spokesman Col. Robert Manning said in a statement.
"Initial assessment indicates that this missile was an intercontinental ballistic missile (ICBM)", he said.
North Korea's state television said the country had achieved its mission of becoming a nuclear state.
North Korea has previously said that its projectiles can hit the US but this marks the first time it says it can do it with this new type of missile, which appears to be an upgraded version of previous models, experts said.
The test, which defied international sanctions imposed over the North Korea's weapons programme, drew swift international condemnation, with the UN Security Council due to convene an emergency session.