In line with the results of the referendum which Montenegro held on 21 May in compliance with the Constitutional Charter of the State Union (of Serbia-Montenegro), I have the honour to ask that Montenegro be admitted to the United Nations, Vujanovic wrote in the letter.
The declaration on independence was enclosed with the letter.
Agence France-Presse (AFP) reported that Podgorica sent letters to all leading world powers and its neighbours on Sunday evening, asking them to recognise Montenegro's sovereignty.
AFP quoted the Montenegrin Foreign Ministry as saying that it sent letters to European Union member-states, UN Security Council member-states, neighbouring countries and leading countries in the world, asking for international recognition.
The ministry also expressed readiness to establish diplomatic relations with those countries.
Montenegro is an independent state with full international legal personality within its existing state borders, reads the declaration the Montenegrin parliament adopted in line with the result of the May 21 referendum at a special session on Saturday evening.
On Monday afternoon, the flag of the State Union of Serbia-Montenegro was removed from a pole in front of the building housing the Serbian Assembly in Belgrade. Serbia's banner was raised instead.
The Montenegrin parliament ceremony was also attended by several parliamentary deputies of the G17 Plus party, led by Finance Minister Mladjan Dinkic, which has advocated Serbia's independence.
Serbia-Montenegro's banners were removed from other places in Belgrade after the Serbian Assembly held a special session on Monday at which it adopted a decision declaring that Serbia was the sole legal successor to the state union of Serbia and Montenegro.