On September 18, 1973, the United Nations admitted the German Democratic Republic and the Federal Republic of Germany, and the two German states sat in the same organisation as separate members for the first time. Both applications had been blocked for 24 years, mostly by the other side's allies, and mostly by the argument that the other state was not legitimate.
The Federal Republic had a strong claim and a weak one. It insisted it was the only legitimate German state, under the Hallstein Doctrine, under which Bonn broke relations with any country that recognised East Germany. That worked until the 1960s, when it became clear that recognising the GDR was what everyone else wanted to do, and that the doctrine mostly isolated Bonn. Willy Brandt, foreign minister and then chancellor from 1969, dropped it. His Ostpolitik was a series of treaties: with the Soviet Union in August 1970 renouncing force, with Poland in December 1970 accepting the Oder-Neisse line, and then the Basic Treaty of December 1972, in which the two German states recognised each other's sovereignty and agreed to exchange permanent missions.
Entry into the UN was the point at which the settlement became irreversible, and it took a legal argument to get there. The Federal Constitutional Court in Karlsruhe ruled in 1973 that the Basic Treaty was compatible with the Basic Law while holding that the German Reich continued to exist and that the constitution still aimed at reunification. That let both states sign the UN Charter with their own declarations. The UN gave both seats. Neither represented the other.
Both were elected to non-permanent positions almost immediately: West Germany served on the Security Council in 1977 and 1978, East Germany in 1980 and 1981. Both joined the specialised agencies. Neither had an army in UN service, and neither was ever going to.
They lasted 17 more years. The Berlin Wall came down on November 9, 1989, and the two Germanies reunited on October 3, 1990, with the GDR acceding to the Federal Republic under Article 23 rather than merging as equals. The German mission in New York is now a single seat, and the two delegations' old buildings are gone.