At a huge military parade that Kim presided over last month, North Korea displayed two of its newest model missiles, including the submarine-launched ballistic type it successfully fired past year and the land-based version it launched last month.
Japan's defense minister says the missile test-fired by North Korea might have been a new type given the altitude and duration of its flight.
North Korea's military programs are shrouded in secrecy and many public displays are filled with props and dummy prototypes prepared just for propaganda.
South Korea's Joint Chiefs of Staff had few other details, including whether the launch was successful or what kind of projectile was sacked Sunday.
A senior North Korean diplomat reportedly says Pyongyang would have dialogue with the U.S. if the conditions were right.
Trump has also said he would be "honoured" to meet North Korean leader Kim Jong Un under the right circumstances.
Experts said Sunday's test showed a considerably longer range than missiles North Korea had previously tested, meaning it had likely made improvements since its February test.
The launch is also the first since a new liberal president took office in South Korea on Wednesday.
Earlier, Japanese officials said the missile landed in the Sea of Japan but outside the country's exclusive economic zone. These medium-ranged missiles would have a reach of between 1,500 and 4,000 kilometers, giving them the ability to strike almost anywhere in the Southeastern Asiatic region.
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"Our military is closely monitoring for provocative movements by North Korea and is maintaining all readiness postures", a statement from the military said.
That missile, launched April 29, blew up over land in North Korean territory, according to a spokesman for the US Pacific Command.
Since the missile has failed several times earlier, the United States doesn't consider another launch to be a crisis at this point, the official said.
Calling the behavior "absolutely unacceptable", Abe said further provocations by North Korea are expected.
David Wright, co-director of the Union of Concerned Scientists, said that the flight time, if true, would mean the missile has a "considerably longer range" than any of the missiles now in the North's arsenal. This missile would be able to reach Seoul, the capital of South Korea, but is unable to reach the southern areas of South Korea.
One of the missiles looked similar to the KN-08 intercontinental ballistic missile that North Korea had included in previous parades.
Choe Son-Hui, a senior official at the North's foreign ministry handling its United States policy, also said Saturday the North would be willing to hold talks with the USA if the conditions are right.
But Jeffrey Lewis, director of the East Asia nonproliferation program at the Middlebury Institute of International Studies at Monterey in California, said at the time that there was an "even chance" that they were intercontinental ballistic missiles (ICBMs).