The test marks a ratcheting up of tensions weeks after a summit between Kim and US President Donald Trump collapsed without agreement.
It also comes after satellite imagery suggested heightened activity at a nuclear test site.
Wednesday's test was "conducted in various modes of firing at different targets" the Korean Central News Agency reported, adding that Kim "guided the test-fire".
The report said Kim described its development as one "of very weighty significance in increasing the combat power of the People's Army".
The "advantages" of the weapon were "the peculiar mode of guiding flight and the load of a powerful warhead", KCNA said.
Its report gave no details of the weapon.
South Korea had not detected anything on radar so it was unlikely to have been a missile, a military official told AFP.
"When North Korea launches a missile, our radar catches it. But no missile was detected," said the official, speaking on condition of anonymity.
Seoul' presidential office said it had no comment.
"The description makes whatever was tested sound like a missile, but that could be everything from a small anti-tank guided missile to a surface-to-air missile to a rocket artillery system," said North Korea analyst Ankit Panda.
Earlier in the week, the Center for Strategic and International Studies, a US monitor, said activity had been detected at Yongbyon, the North's main nuclear testing facility.
The think tank said evidence suggested Pyongyang may be reprocessing radioactive material into bomb fuel.
Kim's Hanoi summit with Trump, the second between the two men, ended abruptly, with North Korea later protesting that the US was being unreasonable in its demands.
Since then, North Korea has said it is mulling options for its diplomacy with the US, and Kim said last week he was open to talks with Trump only if Washington came with the "proper attitude".
"Kim is trying to make a statement to the Trump administration that his military potential is growing by the day," said Harry Kazianis, an analyst at the Washington-based Center for the National Interest.
"His regime is becoming frustrated with Washington's lack of flexibility in recent negotiations."
Koh Yu-hwan, professor of North Korean studies at Dongguk University, agreed the test was a message to the US showing its displeasure over the stalled nuclear talks.
But the fact that it was not a long-range missile or nuclear test "underscores Pyongyang wants to keep alive dialogue with Washington", he added.
"Pyongyang cannot conduct a nuclear or long-range missile launch at this point unless it wants to totally shatter what remains of the US-North talks," he continued.
Pentagon officials said they were aware of the test report but declined to comment further.