North Korean leader Kim Jong Un inspects a Hwasong-12 in an undated photo released by the state-run Korean Central News Agency (KCNA) on Monday. The missile flew 787 kilometers (490 miles) and reached an altitude of 2,111.5 kilometers (1,312 miles).
Analysts say the missile was sacked at an upward trajectory to artificially limit its range during the test.
North Korea's missile program is progressing faster than expected, South Korea's defense minister said on Tuesday, hours after the U.N. Security Council demanded the Kim Jong Un's regime halt all nuclear and ballistic missile tests.
"It appears to have not only demonstrated an intermediate-range ballistic missile (IRBM) that might enable them to reliably strike the USA base at Guam, but more importantly, may represent a substantial advance to developing an intercontinental ballistic missile (ICBM)", it said.
North Korea has said the missile it launched over the weekend was a new type of long-range ballistic rocket that can carry a heavy nuclear warhead.
The statement also condemned an earlier ballistic missile launch by Pyongyang on April 28 - following that launch, Washington began talks with China on possible new United Nations sanctions.
In all, six sets of sanctions have been imposed on North Korea since it first tested an atomic device in 2006.
The Rodong Sinmun said the issue of nuclear weapons is a matter of concern between the North and the US and warned against South Korean interference, accusing the South of being a "puppet" of Washington.
Sunday's test missile flew 787 kilometers (489 miles) across North Korea and into the Sea of Japan, also known as the East Sea, according to state media, and it appeared to have struck near the eastern coast of Russian Federation.More news: 'Life of Kylie': Watch First Teaser for Kylie Jenner's 'Kardashians' Spinoff
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Speaking in Beijing, Mr Putin said nuclear tests of the type that Pyongyang had been carrying out in recent weeks were unacceptable, but that a peaceful solution to rising tensions on the Korean peninsula was needed.
South Korea's military said it could not yet verify the North's claims.
But Melissa Hanham, senior research associate at the James Martin Center for Nonproliferation Studies in California, said the test could just as easily be a stepping stone to a longer-range weapon.
Newly elected President Moon Jae-in said the South had to "learn to say 'no, '" to the USA and to exhaust diplomatic ways of resolving issues with the North.
The North fired a ballistic missile that landed in the sea near Russian Federation on Sunday in a launch that Washington called a message to South Korea, days after its new president took office pledging to engage Pyongyang in dialogue.
The U.S. has always been skeptical that Pyongyang has the technology to mount a nuclear warhead.
A series of pictures published by the hermit state's media showed Jong-Un gazing at the missile in a hangar before the launch.
China is North Korea's largest trading partner and the closest thing it has to an ally, and experts have long said it has the most leverage over the reclusive state.
The statement came after North Korea test-fired a ballistic missile early Sunday, prompting Seoul's new leader, Moon Jae-in, to convene an emergency security meeting.