Nearly 1,700 islanders are eligible to vote and an overwhelming "yes" is expected.
The two-day referendum is being held amid transatlantic tensions that have increased in recent months. It asks: "Do you wish the Falkland Islands to retain their current political status as an Overseas Territory of the United Kingdom?"
Despite the U.K.'s sovereignty over the remote islands, Argentines claim British forces stole the territory from them 180 years ago and have ruled it as a colony ever since. A large majority of the 2,500 inhabitants are of U.K. descent.
Britain and Argentina fought a brief, but bitter, war in 1982 after Argentine troops invaded the Falklands, or Las Malvinas as Argentina calls them. The two months of fighting — ending with Argentina surrendering —resulted in the deaths of 650 Argentines and 255 Britons.
The referendum process is not simply about the vote — it is also being seen as an opportunity for the Falkland Islands government to garner international support.
Britain wants the U.S. In particular to recognize the islanders' rights, but U.S. Secretary of State John Kerry refused to budge during his recent visit to London.
Argentina's government has requested that the U.K. government enter into negotiations to determine sovereignty.
Demands for such talks have increased with the discovery commercially viable oil offshore in the Falklands basin, with pumping due to begin in 2017. That oil means royalties and taxes for the Falklands' government.