The brother of Sinn Fein president Gerry Adams must wait eight days to hear if he can appeal against his extradition to Northern Ireland.
Liam Dominic Adams is wanted by the Police Service of Northern Ireland to stand trial in relation to 18 alleged offences against his daughter Aine Tyrell, who has waived her right to anonymity.
The 56-year-old, who denies the allegations, lost his fight against extradition from the Irish Republic at the High Court in Dublin last Monday.
His barrister Michael O'Higgins applied for leave to appeal against the ruling to the Supreme Court, on five points. Mr Justice John Edwards will give his decision next Tuesday.
Adams, who was supported in court by another daughter, Claire Smith, was granted continuing bail.
Mr O'Higgins argued his client has a constitutional right to have an appeal heard in the Supreme Court. He claimed, if extradited, Adams would be held in prison in Northern Ireland - possibly in solitary confinement for up to 23 hours a day - until his concerns are dealt with by a court.
Aileen Donnelly, senior counsel for the State, said any appeal would be an inappropriate use of resources of the Supreme Court, in terms of both money and time. She said she did not accept Adams' concern about custody was in the public interest.
"There is no acceptance that Mr Adams will be held in 23 hour lock-up," she said.
Arguments that Adams could not get a fair trial because of pre-trial publicity and comments by his brother in support of Ms Tyrell, a delay in bringing charges and changes in the jury selection in Northern Ireland have been rejected by the High Court.
Adams is accused of rape, indecent assault and gross indecency at various addresses in Belfast between March 1977 and March 1983, when the alleged victim was aged between four and 10.