Falklanders will vote in March on whether they wish to maintain their status as a UK overseas territory, in a poll designed to fend off aggressive Argentine claims to the islands Buenos Aires knows as Las Malvinas.
In a Christmas message to the islanders, Mr Cameron restated Britain's commitment to self-determination for the Falklands and made clear he hopes they will vote emphatically to retain their links to the UK.
He accused Argentine president Cristina Kirchner of "misrepresenting" the facts about the Falklands in her campaign to reopen the sovereignty question.
The message came as the Queen appointed a new Governor for the Falklands. Colin Roberts will take up his post at Government House in Port Stanley in April 2014, replacing Nigel Haywood. Mr Roberts said: "I look forward to working with the community and their elected representatives for a secure and prosperous future."
Looking ahead to March, Mr Cameron said: "I value deeply the UK's relationship with the Falklands and hope it will long continue. But it is not my decision, nor is it Argentina's, it is yours and yours alone.
"This referendum is true democracy in action, an opportunity to show the international community what you want for your future and to show it definitively. I hope all of you seize it."