A covering for an architectural-structure covering is disclosed. The covering is formed from a plurality of assembled strips of fabric material joined together. Each assembled strip of fabric material may include a fabric material including inner and outer surfaces, and top and bottom ends. A first slat may be coupled to the inner surface of the fabric material adjacent the top end thereof, the first slat having a first contact surface. A second slat may be coupled to the outer surface of the fabric material adjacent the bottom end thereof, the second slat having a second contact surface. The first contact surface of the first slat of a first assembled strip is preferably nested with the second contact surface of the second slat of an adjoining second assembled strip. The second contact surface of the second slat may be partially covered by the piece of fabric material.