The UK government is due to find out whether it can stop Chagos Islanders returning to their homeland.
The appeal comes after rulings went in the islanders favour
Some 2,000 residents were forced out when the British colony in the Indian Ocean was leased to the US in the 1960s to build an airbase at Diego Garcia.
The government took the case to the Court of Appeal after two earlier rulings declared the actions unlawful.
Many former residents, who have been granted British citizenship, now live in Mauritius or in the UK.
The residents were evicted from their homes on the Chagos archipelago between Africa and Indonesia, in the late 1960s and early 1970s.
The Diego Garcia base, which was crucial during the Cold War, has gained new significance in recent years as a launching point for bombing missions in Iraq and Afghanistan.
In 2000, the courts ruled that Chagossians could return to their homes in 65 of the islands, but not to Diego Garcia. The then Foreign Secretary, Robin Cook, said the government would not appeal.
But in 2004 the government used the royal prerogative - powers that allow action without reference to Parliament - to effectively nullify the decision.
Last year the High Court overturned the order and rejected government argument that the royal prerogative, exercised by ministers in the Queen's name, was immune from scrutiny.
The government took the case to the Court of Appeal, saying the High Court ruling seriously affects the government's control over security matters and its legal relationship with overseas territories.