"'If' the enemy... attempt to use armed forces encroaching upon the sovereignty of the DPRK... the DPRK would use without hesitation all the offensive forces it has possessed, including nuclear weapons," the KCNA news agency quoted him as saying, using the acronym for the North's official name.
Kim was speaking on Wednesday while inspecting a special forces military training base in the west of Pyongyang, KCNA reported.
The remarks came after South Korea staged a military parade earlier this week, with its President Yoon Suk Yeol threatening "the end of the North Korean regime" if Pyongyang used nuclear weapons.
"If North Korea attempts to use nuclear weapons, it will face the resolute and overwhelming response of our military and the US and Republic of Korea alliance," Yoon said.
"That day will be the end of the North Korean regime," he added, addressing thousands of service members gathered at Seoul Air Base for the event.
In response to those remarks, Kim branded the South Korean leader a "puppet" and "an abnormal man", KCNA reported.
Kim's statements also referenced the South's alliance with the United States, which is its principal military partner.
Tens of thousands of US troops are stationed in South Korea.
The South has no nukes of its own and is covered by the US nuclear umbrella.
The latest spat comes weeks after the North disclosed images of a uranium enrichment facility for the first time, showing leader Kim touring the site as he called for more centrifuges to boost the country's nuclear arsenal.
The country, which conducted its first nuclear test in 2006 and is under rafts of UN sanctions for its banned weapons programmes, had never publicly disclosed details of its uranium enrichment facility before.
Relations between North and South Korea are at one of their lowest points in years, with the North recently announcing the deployment of 250 ballistic missile launchers to its southern border.
Pyongyang has designated the South its "principal enemy" and declared itself to be an "irreversible" nuclear weapons power.
North Korea has long flouted UN sanctions thanks in part to support from allies Russia and China.
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