At a news conference in Wilmington, Minner said she would appoint Ted Kaufman to fill the vacancy.
It was not immediately clear if Biden would resign his Senate seat before or after he begins his seventh term in January. However, he must resign before being sworn in Jan. 20 as vice president.
Speculation on Biden's successor had centered in recent weeks on his son, Attorney General Beau Biden. Last week, however, the younger Biden announced that he planned to fulfill his National Guard duties and wouldn't accept an appointment to his father's U.S. Senate seat.