Ian Bailey, a 54-year-old former journalist, denies murdering Sophie Toscan du Plantier, the wife of high-profile film producer Daniel Toscan du Plantier 16 years ago.
Bailey, a former suspect in the Irish investigation into the murder, has been fighting the extradition request from France since April 2010.
French magistrates want to question him over the murder.
At the end of a three-day appeal by Bailey, judges at Ireland's Supreme Court reserved their judgement and did not say when they would deliver their decision.
The Court will pass judgement only on points of law raised by the extradition request, not on a dossier of evidence submitted by Bailey's lawyers which alleges the Irish probe into the murder was seriously flawed.
Bailey lived just a few kilometres from Sophie Toscan du Plantier's home in Schull, County Cork, southern Ireland. She was found beaten to death near the house on December 23rd, 1996. She was 39.
Suspicions about Bailey's involvement were aroused when he was one of the first people to arrive at the scene of the murder.
Investigators say that in articles he wrote about the case, he included details which only police and the murderer could have known.