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.
Cristina Fernandez de Kirchner has reignited the row over the future of the Falkland Islands
"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 foreign office said they would not negotiate on the sovereignty of the Falklands
"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 letter as published in the Guardian
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.