On September 17, 1991, the United Nations admitted North and South Korea together, as two separate member states, by acclamation and without objection. Each had spent 40 years insisting it was the only legitimate government on the peninsula and that the other was a fiction. Neither got a veto, because both had agreed not to try to block the other.
Money drove the reversal. North Korea had opposed separate membership all through the 1970s and 1980s, and had proposed a single seat under a confederal name. What changed was that the Soviet Union, which had been the main backer of that position, normalised relations with Seoul in September 1990 and then collapsed. Moscow could no longer be relied on to block anything, and Pyongyang calculated that being inside the UN was better than being outside it while the South was in.
Both applied in December 1990. The applications were held up while the Security Council argued about whether the General Assembly had the authority, and South Korea, which had wanted in for years, was careful not to press for a vote that would look like a victory. The two governments had also signed an accord on reconciliation in December 1991, in which each agreed to respect the other's system, and a joint declaration on denuclearisation of the peninsula.
The same day brought in seven other members: Estonia, Latvia and Lithuania, the Marshall Islands and Micronesia, and the two states that are now the Czech Republic and Slovakia as part of Czechoslovakia. It was the largest single expansion since 1955, and it reflected the end of the Cold War more than any speech did.
It settled nothing. The two states have no diplomatic relations, the armistice of 1953 has never been replaced by a treaty, and both still formally claim the whole peninsula. North Korea has tested nuclear weapons since 2006 and is under UN sanctions for it. South Korea joined the OECD in 1996 and is one of the world's largest economies. The two flags stand next to each other in New York and have since 1991.