Fashion retailer Primark has signed a long-term lease for 112,000 square feet at the Burnham Building in the Millennium Tower complex at Boston’s Downtown Crossing.

Dublin-based Primark’s first U.S. store will occupy four floors in the eight-story Burnham Building, including 70,000 square feet of retail floor space.

Cushman & Wakefield represented developer Millennium Partners in its negotiations with Primark, which has 271 stores in the U.K., Ireland and Europe.

The Burnham Building was completed in 1912 for Filene’s Department Store. Renovations began in summer 2013 and are scheduled to be completed next year.

Ad agencies Havas Media and Arnold Worldwide agreed last April to lease 125,000 square feet in the Burnham Building, and Wellesley-based grocery chain Roche Bros. announced in September that it would open a 37,000-square-foot supermarket. The Burnham Building retail space is now 97 percent leased, with only a 1,900-square-foot corner storefront available.

The $689 million Millennium project is anchored by a 625-foot tower that will contain 95,000 square feet of retail space and 442 luxury condos.