On June 6, Israeli Foreign Minister Abba Eban travelled to New York to forestall a UN Security Council cease-fire. There he met Ambassador Gideon Rafael who had been fighting the passage of a cease-fire resolution that, according to Michael Oren in Six Days of War, "restored the status quo ante bellum but without ending the status belli."
A reworked draft resolution was passed later in the day, and Israel and Jordan accepted it while Egypt, Syria and Iraq rejected it.
On the third day of war, with Israel beginning to seize territory in Jerusalem, the Jordanians beseeched the UN and the Americans to persuade Israel to accept a cease-fire leaving the Old City of Jerusalem in Jordanian hands. Israelis responded to the Jordanian message with an ultimatum in the form of a question to King Hussein: Was he really in charge of his armed forces? As evidence, Israel asked that Hussein provide an exact time when the shelling on the western part of the city would cease and requested that he determine a meeting place for both sides to discuss a cease-fire and permanent settlement. The possibility of a Security Council cease-fire prompted an Israeli approach towards the Old City to reach the Jewish holy sites as quickly as possible. Jordanís response was evasive, and Israel went ahead with its attack.
On the fourth day of war, June 8, Nasser and other Egyptian officials initially reiterated Egyptís rejection of any cease-fire. But when Israel reached the Suez Canal, and as the decimation of Egyptís army could not longer be covered up and as foreign press reports finally penetrated the country, Nasser later that day instructed his bewildered ambassador to accept a cease-fire.
On 9 June, Syria announced its acceptance of the cease-fire if Israel were to honor it. But Israel, loathe to leave the Syrians on the strategic Golan Heights from which they had attacked Israelis for so long, later that day launched an attack on Syria. The next day, Syria demanded an emergency session of the Security Council, which resulted in a statement reaffirming the councilís desire for a cease-fire and advising U Thant to consult the countries at war.
The cease-fire on the Golan Heights went into effect on June 10 and remained until Oct. 6, 1973.
Atlas of the Arab-Israeli Conflict, Martin Gilbert, 1993
Six Days of War: June 1967 and the Making of the Modern Middle East, Michael B. Oren, 2002