The long-awaited appeal by Stephen Downing is being heard by three judges in London.
Downing spent 27 years in prison for a murder he says he did not commit.
The 45-year-old was just 17 when he was arrested for the murder of Wendy Sewell in his home town of Bakewell, Derbyshire, in 1973.
Last February, he was released on bail pending appeal and returned to live with his parents, Ray and Juanita, and sister Christine.
He could receive more than €1.27m in compensation if, as expected, his appeal succeeds.
Bail was not opposed by the British Crown, which conceded last year that the appeal was "highly likely" to succeed in the light of serious questions raised over the admissibility of Downing's initial confession statements, which formed a main plank of the prosecution case.
The judge who granted bail said that, having read the papers in the case, he agreed with British Crown lawyers that the conviction would probably be quashed on appeal - although "the court may ultimately take a different view".
The murder victim's badly-beaten body was found by Downing in a cemetery where he worked.
Since the trial, he has consistently denied murder and so has been ineligible for parole, which could have freed him from his life sentence in 1990.
The Criminal Cases Review Commission referred his case to the Court of Appeal after years of campaigning by his parents and by Don Hale, editor of the Matlock Mercury.