The US envoy to the UN has urged the Security Council to take the "strongest possible measures" against North Korea after its latest nuclear test.
The US will circulate a new draft Security Council resolution, hoping for a vote next Monday.
South Korea is set to scrap a warhead weight limit on its own missiles.
President Moon Jae-in agreed in a phone call with his US counterpart, Donald Trump, to remove the current cap of 500kg (1,100lb), the South Korean presidential office said.
It carried out live-fire exercises on Monday, simulating an attack on the North's nuclear test site.
Reports suggest the North is preparing new test missile launches. It tested a nuclear bomb underground on Sunday. Estimates of its power range from 50 kilotonnes to 120 kilotonnes. A 50kt device would be about three times the size of the bomb that destroyed Hiroshima in 1945.
Opening Monday's meeting, UN Under Secretary General Jeffrey Feltman said North Korea's actions were destabilising global security, and he called on Pyongyang to abide by Security Council resolutions.
"The DPRK [North Korea] is the only country that continues to break the norm against nuclear test explosions," he said.