The letter, published as an advert in the Guardian and reported in other newspapers, says that Argentina was forcibly stripped of the Malvinas - the Argentinian name for the islands - in "a blatant exercise of 19th-century colonialism".
The 59-year-old president, who made several calls for the return of the islands during last year's 30th anniversary of the two countries going to war, urged the Prime Minister to abide by United Nations resolutions she says back the Argentinian cause.
"One hundred and eighty years ago on the same date, January 3rd, in a blatant exercise of 19th-century colonialism, Argentina was forcibly stripped of the Malvinas Islands, which are situated 14,000km (8700 miles) away from London," she said in the letter, copied to United Nations secretary general Ban Ki-moon.
"The Argentines on the Islands were expelled by the Royal Navy and the United Kingdom subsequently began a population implantation process similar to that applied to other territories under colonial rule.
"Since then, Britain, the colonial power, has refused to return the territories to the Argentine Republic, thus preventing it from restoring its territorial integrity.
"The Question of the Malvinas Islands is also a cause embraced by Latin America and by a vast majority of peoples and governments around the world that reject colonialism."
Mr Cameron and Ms de Kirchner clashed over the Falklands when the pair came face to face at the G20 summit in Mexico last June.
He rejected her demand for negotiations over the sovereignty of the islands and told her that she should respect the result of a referendum next year, when the Falklanders will vote on whether they wish to retain their ties with Britain.
The Argentine president had earlier taken her demands to the United Nations, appearing at the annual meeting of the little-known UN Decolonisation Committee on the 30th anniversary of Britain's ousting of an Argentinian invasion force from the Falklands.
She used the occasion to reiterate Argentina's opposition to any more wars and to criticise the Prime Minister's decision to mark the day by flying the Falklands flag over his official 10 Downing Street residence.
In December Argentina protested at Britain's decision to name a vast swathe of Antarctica Queen Elizabeth Land, with its foreign ministry handed a formal protest note to British ambassador John Freeman in Buenos Aires.
The area, which makes up around a third of the British Antarctic Territory, is also claimed by the South American country.
Ms de Kirchner's letter continues: "In 1960, the United Nations proclaimed the necessity of 'bringing to an end colonialism in all its forms and manifestations'.
"In 1965, the General Assembly adopted, with no votes against (not even by the United Kingdom), a resolution considering the Malvinas Islands a colonial case and inviting the two countries to negotiate a solution to the sovereignty dispute between them.
"This was followed by many other resolutions to that effect.
"In the name of the Argentine people, I reiterate our invitation for us to abide by the resolutions of the United Nations."
A Foreign Office spokeswoman said that the Falkland Islanders "are British and have chosen to be so".
"They remain free to choose their own futures, both politically and economically, and have a right to self-determination as enshrined in the UN Charter," she added.
"This is a fundamental human right for all peoples. There are three parties to this debate, not just two as Argentina likes to pretend.
"The islanders can't just be written out of history. As such, there can be no negotiations on the sovereignty of the Falkland Islands unless and until such time as the islanders so wish."
Last week newly released secret documents revealed Britain considered pulling out of the 1982 World Cup as war broke out with Argentina over the Falkland Islands, with sports governing bodies urged to pull out of competitions with Argentinian teams.
Argentina's President Cristina Fernandez
Argentina's President Cristina Fernandez de Kirchner listens during a meeting at the United Nations headquarters in New York on the disputed Falkland Islands on the 30th anniversary of the end of war between the Britain and Argentina, on June 14, 2012. Argentina has said Britain has 'a duty' to negotiate the future of the Falkland Islands. Kirchner leads delegation of more than 90 Argentine diplomats and officials at the UN decolonization committee's annual meeting on the Falklands and 15 other territories around the world. Two Falkland Islands assembly members will put the case that the 3,000 Falklanders want to remain under the British flag. AFP PHOTO/Mehdi Taamallah (Photo credit should read MEHDI TAAMALLAH/AFP/GettyImages)
Picture of Port Stanley, in the Falkland
Picture of Port Stanley, in the Falkland Islands, taken on March 29, 2012. Next April 2 marks the 30th anniversary of the war between Britain and Argentina for the possession of the islands. AFP PHOTO/MARTIN BERNETTI (Photo credit should read MARTIN BERNETTI/AFP/GettyImages)
Argentinian Marcelo Wytrykusz, veteran o
Argentinian Marcelo Wytrykusz, veteran of the 1982 South Atlantic war between Argentina and Britain over the Falkland Islands (Malvinas), remains chained to the fence of the building of Argentine oil company YPF, in Buenos Aires, on June 5, 2012. Wytrykusz, member of the National Resistance movement, is protesting against the contracting -by recently nationalized YPF- of the 'Stena Polaris', a ship of British, flag for the transportation of oil from Tierra del Fuego to Buenos Aires. AFP PHOTO/ALEJANDRO PAGNI (Photo credit should read ALEJANDRO PAGNI/AFP/GettyImages)
A map of the Falkland Islands is inscri
A map of the Falkland Islands is inscribed in stone during the dedication ceremony of the Falklands Memorial at the National Memorial Arboretum in Alrewas, central England on May 20, 2012. More than 600 veterans and family members gathered to commemorate the 30th Anniversary of the Task Force Landings on the Falkland Islands. Initiated by The South Atlantic Medal Association 1982, the Falklands Memorial has been built to honour the Task Force and to remember the 255 UK servicemen and merchant seamen who gave their lives in the Falklands Conflict. AFP PHOTO/PAUL ELLIS (Photo credit should read PAUL ELLIS/AFP/GettyImages)
Tour guide Derek Pettersson shows the wr
Tour guide Derek Pettersson shows the wreckage of an Argentine trench, used during the 1982 conflict, near Port Stanley, in the Falkland Islands, on March 29, 2012. Next April 2 marks the 30th anniversary of the war between Britain and Argentina for the possession of the islands. AFP PHOTO / MARTIN BERNETTI (Photo credit should read MARTIN BERNETTI/AFP/GettyImages)
View of the St Mary's Church in Port Sta
View of the St Mary's Church in Port Stanley, in the Falkland Islands, on March 28, 2012. April 2, 2012 commemorates the 30th anniversary of the war between Britain and Argentina for the possession of the islands. AFP PHOTO / MARTIN BERNETTI (Photo credit should read MARTIN BERNETTI/AFP/GettyImages)
The flag of the Falkland Islands flies o
The flag of the Falkland Islands flies over number 10 Downing Street in central London on June 14, 2012. Britain's premier vowed on June 14 to defend the Falklands from Argentinian 'aggressive threats' as the 30th anniversary of the end of the war over the islands was marked in London and Port Stanley. AFP PHOTO / LEON NEAL (Photo credit should read LEON NEAL/AFP/GettyImages)
Argentinian Marcelo Wytrykusz, veteran o
Argentinian Marcelo Wytrykusz, veteran of the 1982 South Atlantic war between Argentina and Britain over the Falkland Islands (Malvinas), remains chained to the fence of the building of Argentine oil company YPF, as a British flag is burnt in Buenos Aires, on June 5, 2012. Wytrykusz, member of the National Resistance movement, is protesting against the contracting -by recently nationalized YPF- of the 'Stena Polaris', a ship of British, flag for the transportation of oil from Tierra del Fuego to Buenos Aires. AFP PHOTO/ALEJANDRO PAGNI (Photo credit should read ALEJANDRO PAGNI/AFP/GettyImages)
Argentine President Cristina Fernandez d
Argentine President Cristina Fernandez de Kirchner delivers a speech during a ceremony to mark the 30th Anniversary of the 1982 South Atlantic war between Argentina and the Britain over the Falkland Islands (Malvinas), in Ushuaia, Tierra del Fuego, some 3100 km south of Buenos Aires, Argentina on April 2, 2012. Britain and Argentina on Monday marked 30 years since an Argentine invasion of the Falklands Islands triggered a bloody 74-day war, amid renewed tensions between the two countries. AFP PHOTO/JUAN MABROMATA (Photo credit should read JUAN MABROMATA/AFP/Getty Images)