Ban, a former South Korean foreign minister, expressed hope that there will be no increase of tensions on the Korean peninsula following the execution of Jan Song Thaek, widely considered the No. 2 official in North Korea.
Ban, making his first public comments on the execution, also urged North Korea's leaders to work to denuclearise the Korean peninsula.
He stressed that North Korea must focus on improving the living conditions of its people and comply with Security Council resolutions, which it has long ignored.
Many analysts said the acknowledgement of dissension and dangerous instability in the government is an alarming prospect as Kim tries to revive a moribund economy while maintaining and advancing development of the country's nuclear arsenal.
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Ban stressed that the United Nations has a longstanding and very clear position against the death penalty "under any circumstances."
He appealed to "all the parties concerned surrounding the Korean peninsula, while they must be diligently and carefully watching the development of situation, not to take any premature actions."