Last year Pyongyang carried out 20 ballistic missile tests, almost all of them in the early hours of the morning.
Moon routinely summons his National Security Council immediately after such events, and the timing has ensured officials, diplomats and journalists in Seoul have regularly been jolted awake.
But when he met Seoul's envoys this week a jovial Kim pledged not to disrupt Moon's slumbers any more.
"I've made up my mind today and President Moon does not have to be disturbed from sleep with early morning wake-up calls," he was quoted as saying by an official of Seoul's presidential Blue House.
At the four-hour meeting on Monday, Kim agreed to hold a summit with Moon in April, expressed his desire to meet US President Donald Trump at an early date, offered to to put denuclearisation on the table, and promised no more missile or nuclear tests while dialogue continued.
The two Koreas also agreed to open a hotline between the leaders.
"If things don't go well with officials' talks and they behave arrogantly, Mr President and I can now talk directly through the phone and sort things out easily," Kim said, to laughter from delegates on both sides.
Kim, whose weight is a regular subject of comment in foreign media, was aware of how he is portrayed overseas and engaged in pleasantries bordering on self-deprecation, another Blue House official said according to reports.
No details were provided. But Trump once labelled him as "short and fat".
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