Bargains will be back at Boston's Burnham Building, thanks to a fast-growing European retailer making its first inroads in the U.S.

Primark, a clothing chain whose formula of fashionable looks at rock-bottom prices has proved a hit with U.K. shoppers, said Wednesday it plans to open its first U.S. store late next year in the former Boston home of the original Filene's Department Store.

"After...