There was no immediate independent confirmation of the test that was touted by Kim as a "miraculous achievement" of his country.
He described the missile as "time bomb which will go off on the backs of our hostile enemies at any time," according to the KCNA, which reported that the missile was launched from a submarine far from the mainland.
The state press did not specify when or where the launch was carried out, and also kept mum on the size and range of the missile.
The announcement of the launch came a day after Pyongyang threatened to open fire without warning on any South Korean naval ships entering the countries' disputed Yellow Sea border off the west coast.
Last month, the head of US Northern Command Admiral William Gortney told reporters that North Korea may be capable of putting a nuclear warhead on a long-range missile that could reach the United States.
North Korea declared itself a nuclear power in 2005 and conducted a series of underground nuclear weapon tests in 2006, 2009 and 2013. Pyongyang's nuclear program has triggered multiple protests from the international community.