Let T be the unit circle,(mu) be a Borel probability measure on T and (phi) be a bounded Lebesgue measurable function on T. in this paper we consider the weighted composition operator W(phi) on L^2(T,mu) defined by W(phi)f:=(phi)*(f(circle)(tau)), f in L^2(T), where (tau) is the map (tau)(z)=z^2, z in T. We will study the von Neumann-Wold decomposition of W(phi) when W(phi) is an isometry and (mu)<< m,where m is the normalized Lebesgue measure on T.