It is the first such case to be considered by the Court of Criminal Appeal in Edinburgh since a change in the law relaxed the "double jeopardy" rule.
The legal debate is expected to last two weeks - but details cannot be reported.
Lord Justice Clerk Lord Carloway, sitting with Lady Dorrian and Lord Bracadale will eventually rule on whether the earlier acquittal of 68-year-old Angus Sinclair should be quashed - paving the way for a new trial.
The bodies of Christine Eadie and Helen Scott, both 17, were found in East Lothian after they had been seen drinking in the World's End pub in the Royal Mile, Edinburgh.
Sinclair faced charges of rape and murder until the September 2007 trial was brought to an end when judge Lord Clarke ruled there was no case to answer.
Since then the Scottish Parliament has passed the Double Jeopardy (Scotland) Act 2011.
Lord Advocate Frank Mulholland, QC - who is ultimately responsible for all prosecutions in Scotland - has used the Act to bring the case back to court.
Before the legal arguments began yesterday he told the appeal judges he was aware of media interest because this was the first application made under the Act.
Defence QC Ian Duguid, for Sinclair, is opposing the retrial move.
A decision is expected, in writing, at a later date.