The Grothendieck compactness principle states that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. In Dowling et al. (J Funct Anal 263(5):1378–1381, 2012), an analogue of the Grothendieck compactness principle for the weak topology was used to characterize Banach spaces with the Schur property. Using a different analogue of the Grothendieck compactness principle for the weak topology, a characterization of the Banach spaces with a symmetric basis that are not isomorphic to \(\ell ^1\) and do not contain a subspace isomorphic to \(c_0\) is given. As a corollary, it is shown that, in the Lorentz space \(d(w,1)\), every weakly compact set is contained in the closed convex hull of the rearrangement invariant hull of a norm null sequence.