In a preferred embodiment, a continuous elastic ribbon is fed to a diaper web assembly station in a stretched condition. While traveling to the assembly station, an adhesive is applied to discrete lengths of the elastic ribbon at regularly spaced intervals. Simultaneously, discrete absorbent core segments as well as webs of moisture-impervious backsheet material and moisture-pervious topsheet material are fed to the diaper web assembly station. At the station, the stretched elastic ribbon is adhered to the moisture-impervious backsheet web in the discrete areas of the elastic which are covered by adhesive at predetermined points along the length of said web. After the adhesive has set up, the assembled web and the elastic contained therein is severed in its unadhered areas, whereupon the unadhered end portions become relaxed and inactive without affecting the functionality of the adhered portions in the ultimate assemblage. The inventive concept in its broadest sense relates to the continuous adherence of discrete lengths of stretched elastic to a continuously moving web at relativey high operating speeds and thereafter severing the elastic in the unadhered areas to produce elasticized structures having discrete strands of elastic adhered thereto at predetermined locations along their length. In a particularly preferred embodiment, the aforementioned process is utilized to apply discrete elastic leg bands to a disposable diaper structure to provide improved containment and fit therein.