Bishop Quinn, who has headed the Diocese of Winona-Rochester for 13 years, is 76. Canon law requires bishops to submit their resignation to the pope when they turn 75.
Bishop Barron, 62, is a Chicago native who has been a Los Angeles auxiliary bishop since 2015. He also is the founder of Word on Fire Catholic Ministries.
The changes were announced in Washington June 2 by Archbishop Christophe Pierre, apostolic nuncio to the U.S.
The Diocese of Winona-Rochester comprises 20 counties in southern Minnesota and has a Catholic population of 132,825.