Angus Sinclair stood trial more than five years ago, accused of murdering 17-year-olds Christine Eadie and Helen Scott. He was acquitted of the charges, which he had denied.
The two girls were last seen leaving Edinburgh's World's End bar in 1977.
Prosecutors want to try Sinclair again for the murders and have applied to the High Court for his acquittal to be "set aside".
The move follows changes made in Scotland in 2011 to the centuries-old double-jeopardy principle, which prevented a person being tried twice for the same crime.
The Crown began court proceedings in the case today, with Lord Advocate Frank Mulholland QC addressing three judges at the Appeal Court in Edinburgh on procedural issues, including the state of preparation for future hearings.
A further procedural hearing has been scheduled for May 14.
Defence QC Ian Duguid, representing Sinclair, told the court today: "As far as the application is concerned, I can indicate that, after discussion with the respondent (Sinclair), the application is opposed."
Sinclair was not present in court today due to a "medical condition".
Further court sittings could see judges hearing some of the evidence which prosecutors would intend to lead in any future retrial. Mr Duguid indicated that the defence plans to investigate whether the evidence is in fact new.
In a Scottish legal first, prosecutors indicated in December that they had applied to the High Court for authority under double jeopardy legislation to set aside the acquittal and prosecute Sinclair again.
It was the first application to be made since the Double Jeopardy (Scotland) Act came into force in November 2011.