A treaty banning nuclear tests in the atmosphere, in outer space and under water came into force on October 10, 1963, two months after the United States, the Soviet Union and Britain signed it in Moscow. Underground tests were left alone, provided no fallout crossed a frontier. It was the first arms agreement of the nuclear age that actually stopped something: within a year the three powers that had been exploding devices over the Pacific and over Novaya Zemlya had stopped.
What moved it was fallout. By the late 1950s the strontium-90 from atmospheric tests was being measured in milk, in the teeth of children and in the soil of every country that kept dairy herds, and a group of American and Soviet scientists had been arguing since 1957 that a ban could be checked by seismology, because an earthquake and an explosion leave different traces on a seismograph and nobody needs to be let in. Verification settled it. The negotiations had been stuck for years on the Western demand for inspectors and the Soviet refusal to have them.
Fright helped the politics along. Kennedy and Khrushchev had come as close to nuclear war as anyone has, over Cuba in October 1962, and both concluded afterwards that something had to slow the arms race. Kennedy proposed the treaty in a speech at American University in June 1963. The text was agreed in Moscow in twelve days. It was opened to everyone and more than 100 states signed.
What it did not do was end testing. Two powers stayed out: China carried out its first atmospheric test on October 16, 1964, and France tested in the atmosphere until 1974, both arguing that an agreement reached between the superpowers was not theirs to join. Underground testing rose, since that was the gap the text left. It did lead on to the Non-Proliferation Treaty of 1968 and the Comprehensive Test Ban Treaty of 1996, which has still not come into force.