Mueller, 65, replaces U.S. Cardinal William Joseph Levada as prefect of the congregation for the doctrine of the faith, ANSA reported Monday.
The post has come under fire recently for the way it has handled the worldwide sexual abuse scandal, the news service said.
Levada, 76, who had replaced the pope -- then Cardinal Joseph Ratzinger -- when he left the position, resigned from the post last year.
Mueller also will handle negotiations with the breakaway Lefebvrian ultra-traditionalists who are being brought back into the church. He will also oversee the International Theological Commission and the Biblical Commission, ANSA reported.