The settlement comes as the cardinals of the Catholic Church are meeting to select a new pope following the abdication of Benedict XVI.
Cardinal Mahony is at the conclave.
Nearly 10,000 US Catholics signed a petition urging him not to go because of the allegations he protected priests accused of child sex abuse.
Cardinal Mahony was the popular archbishop of the Los Angeles Archdiocese - the largest in the US, representing five million Catholics - for 26 years until his retirement in 2011.
In this case, he is accused of helping a confessed paedophile priest evade the law by sending him away to a church-run treatment centre before returning him to ministry in LA.
The priest went on to abuse again and this payment of $9.9m relates to four child sex abuse cases.