Liam Adams, 56, was detained after the High Court ruled, the Belfast Telegraph reported. The extradition order takes effect in 15 days.
Aine Tyrell, who went public in a 2009 television documentary, says her father sexually abused her for six years, starting in 1977 when she was 4.
Liam Adams was arrested in 2010 in Ireland on a European arrest warrant issued by the Police Service of Northern Ireland. He had left Northern Ireland, saying he would not get a fair hearing.
Another daughter, Claire Smith, accompanied Adams at the High Court hearing. He embraced her as he was taken into custody.
Gerry Adams, the longtime president of the Sinn Fein party, was elected to the Dail, the Irish parliament, after serving as an abstentionist member of the British Parliament.