An elastic composite formed from a laminate that contains an elastic film and a lightweight nonwoven facing having a low strength in the cross-machine direction ("CD") is provided. Due to the low strength of the facing, it is desirable that the elastic film have a sufficient thickness and weight to enhance the strength of the resulting composite. Unfortunately, the ability to join an elastic film to such a lightweight nonwoven facing becomes increasingly difficult as the thickness of the film increases. In this regard, the present inventors have discovered that a second laminate may be employed in the elastic composite that is formed from a thermoplastic film and nonwoven facing to impart increased strength to the composite. The film and facing of the second laminate may be formed from the same or different materials than the film and facing of the first laminate. Regardless, the laminates are positioned so that the elastic and thermoplastic films are in a face-to-face relationship. Through selective control over the polymer content and thickness, the present inventors have discovered that the elastic and thermoplastic films may be readily joined together under light pressure, even at ambient temperature conditions. This avoids the application of a substantial amount of heat used in conventional calender bonding processes, which can otherwise damage the film, and also minimize the need for costly adhesives.