It said the test was successful, but did not specify when or where it took place.
South Korea's Joint Chiefs of Staff said it detected two projectiles fired late on Saturday, which flew 110 kilometres (68 miles) at an altitude of 25 kilometres, travelling at speeds of around Mach 4.
The United States was "aware of the North Korean statement that they conducted a test of a long range artillery system", a Pentagon spokesperson said, adding it was monitoring.
Analysts had widely expected Pyongyang could conduct a nuclear test as part of events to celebrate Friday's anniversary of the 110th birthday of North Korea's founding leader -- and Kim's grandfather -- Kim Il Sung.
Expectations were heightened because of indications that Pyongyang had restarted work at one of its known nuclear testing sites.
Analysts said the weapon tested over the weekend appeared to be a new short-range ballistic missile -- but no less significant.
"This is North Korea's first tactical nuclear weapon delivery system, it would seem," said Ankit Panda, a senior fellow at the Carnegie Endowment for International Peace.
"You don't have to be particularly imaginative to put this two and two together."
Short-range but significant
Photos carried by the Rodong Sinmun newspaper showed a grinning Kim -- surrounded by uniformed officials -- applauding as he watched what it said was the test-firing of the weapon.
Kim gave a military research team "important instructions on further building up the defence capabilities and nuclear combat forces," according to the KCNA report.
Kim had also overseen in Pyongyang a series of events to celebrate Friday's anniversary, including a huge civilian parade, mass dance parties, and fireworks -- but no military parade to show off the regime's new weapons.
At a key party congress in January 2021, Kim outlined a five-year defence development plan, calling for the development of higher nuclear technology and to make smaller and lighter nuclear weapons for "more tactical uses."
The remarks signalled a return to nuclear testing and Kim's rare attendance at a launch of a short-range missile -- the first in more than two years -- suggested "particular significance", according to Panda.
North Korea had paused long-range and nuclear tests while Kim met then-US president Donald Trump for a bout of doomed diplomacy, which collapsed in 2019.
North Korea has tested nuclear weapons six times since 2006 and touted the success of its last and most powerful one in 2017 -- a hydrogen bomb with an estimated yield of 250 kilotons.
Officials and analysts had widely predicted that North Korea may carry out its seventh nuclear test in the coming weeks.
Satellite imagery has shown signs of new activity at a tunnel at the Punggye-ri nuclear testing site, which North Korea said was demolished in 2018 ahead of the first Trump-Kim summit.
"This test of a tactical nuclear delivery system comes as indicators grow of significant reconstitutive work at North Korea's Punggye-ri nuclear test site," said Panda.
With a new nuclear test, experts say Pyongyang will seek to miniaturise nuclear warheads to mount on its ICBMs.
South Korean officials have said Pyongyang could still stage a military parade or carry out a weapons test on or around 25 April, the anniversary of the founding of the Korean People's Army.
South Korea and the United States regularly stage military exercises, but Pyongyang has long protested the drills as a rehearsal for war.
"This training is a defensive command post training using computer simulation, and there is no real military maneuver training," Seoul's Joint Chiefs of Staff said Sunday.