The test had provided a "sure guarantee for the reliability of the inter-continental ballistic missile (ICBM) warhead re-entry," Kim was quoted as saying by the North's official KCNA news agency on Tuesday.
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A proven ability to build a warhead capable of standing the intense heat and vibration of atmospheric re-entry would be a major step-up in North Korea's nuclear strike capability, comprising a clear threat to the US mainland.
The latest claim follows more than a week of dire threats and warnings by Pyongyang of strikes against Seoul and Washington over ongoing South Korea-US large scale military exercises.
Kim's comments also came days after he was photographed posing with what state media said was a miniaturised nuclear warhead capable of fitting on a ballistic missile.
In order to boost the reliability of the country's strike capability, Kim said a nuclear warhead explosion and test firings of "several kinds" of ballistic rockets "will be conducted in a short time."
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Kim had said last week that the North was planning to conduct new tests to gauge the "destructive power" of the new, miniaturised warheads.
While North Korea is known to have a small stockpile of nuclear weapons, its ability to deliver them accurately to a chosen target on the tip of a ballistic missile has been a subject of heated debate.
There are numerous question marks over the North's weapons delivery systems, with many experts believing it is still years from developing a working ICBM that could strike the continental United States.
Others have doubted whether any miniaturised device the North has designed would be robust enough to survive the shock, vibration and temperature change associated with ballistic flight.
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And most experts rule out the prospect of North Korea launching any sort of nuclear strike with a largely untested system, saying it would be tantamount to suicide given overwhelming US technical superiority.
Tensions have been rising on the Korean peninsula ever since the North conducted its fourth underground nuclear test on January 6, followed by a long-range rocket launch a month later.
The UN Security Council responded by adopting a resolution earlier this month laying out the toughest sanctions imposed on Pyongyang to date over its nuclear weapons programme.
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The resolution broke new ground by targeting specific sectors key to the North Korean economy and seeking to undermine the North's use of, and access to, international transport systems.
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