Osbourne announced last week that she would not be back when the new season of the show begins later this year.
Cole -- a member of the girl group Girls Aloud, which won the reality competition show "Popstars: The Rivals" -- will join Simon Cowell, Louis Walsh and Dannii Minogue on "The X Factor," the BBC said.
"It's a massive honor to be following in the footsteps of Sharon Osbourne," Cole told the BBC.
"The rest of the girls have said they're right behind me, which is really important for me as it will be weird to be on the other side of the fence this time. So whilst we get started on the next album, it will be brilliant to be a judge on 'The X Factor.'"