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In this paper we investigate the ell^p space structure of theBanach envelope of operatorname{Weak}L_1. In particular, theBanach envelope of operatorname{Weak}L_1 contains a complementedBanach sublattice that is isometrically isomorphic to thenonseparable Banach lattice ell^p, (1 leq p <infty) as wellas the separable case.