An elastic laminate is made by first coextruding a film having an elastic outer film layer formed of a styrene-block copolymer and an inner film layer of lesser elasticity than the outer layer. An outer fleece web with an outer adhesive layer is then adhered to the outer film layer, and an inner fleece web with an inner adhesive layer is adhered to the inner film layer so as to form between the inner fleece web and the inner film layer a bond that is weaker than a bond formed by the outer adhesive layer between the outer fleece web and the outer film layer. Thereafter the coextruded film bonded by the inner and outer adhesive layers to the outer and inner fleece webs is prestretched such that the weak bond formed by the inner adhesive layer separates only locally from the inner film layer but the bond formed by the outer adhesive layer remains intact.