In a legal first in Scotland, prosecutors want to try Angus Sinclair again for the murders and have applied to the courts for his 2007 acquittal to be set aside.
He had been accused of murdering 17-year-olds Christine Eadie and Helen Scott, but denied the charges.
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The two girls were last seen leaving Edinburgh's World's End bar in 1977 and their bodies were found in separate locations in East Lothian the following day.
The bid follows changes made to the centuries-old double jeopardy principle, which prevented a person being tried twice for the same crime.
The Crown yesterday began new proceedings at the Appeal Court in Edinburgh, with Lord Advocate Frank Mulholland, QC, Scotland's top prosecutor, addressing three judges on procedural issues.
A further procedural hearing has been scheduled for May 14.
Defence QC Ian Duguid, representing Mr Sinclair, said: "As far as the application is concerned, I can indicate that, after discussion with the respondent [Mr Sinclair], the application is opposed."
He said the defence wanted to look into whether what was being put forward "constitutes new evidence at all".
Further court sittings could see judges hearing some of the evidence which prosecutors would intend to lead in any future retrial.
Mr Sinclair was not present for the court hearing, before Lord Carloway, sitting with Lord Menzies and Lady Smith, due to a medical condition.
Mr Mulholland said it is decades since the killings and the families were anxious to have finality in the matter, with the Crown ready to proceed to a full hearing on the issue.
The Lord Advocate said: "There are tests, hurdles, which require to be overcome."
He said that involved the court considering whether the case was substantially strengthened by new evidence and whether a reasonable jury would convict on the basis of new evidence and previous material.
Mr Mulholland said it was estimated eight days would be sufficient for a hearing on the Crown application to set aside the acquittal.
Mr Sinclair's trial for the murders collapsed in 2007 after Lord Clarke ruled there was insufficient evidence for the case to go to the jury.
A second prosecution can be launched under new double jeopardy laws if a strict set of criteria are met, including whether new evidence or a confession comes to light.
This is just one of a number of cases being considered by the Crown to revisit under the new laws.
Others include the murders of Indian waiter Surjit Singh Chhokar in Overtown, North Lanarkshire, in 1998, and drama student Amanda Duffy in Hamilton in 1992. Both cases are currently being looked at by police.