The space WeakL 1 consists of all Lebesgue measurable functions on [0,1] such thatq(f)=supcλ{t:|f(t)|>c} c>0 is finite, where λ denotes Lebesgue measure. Let ρ be the gauge functional of the convex hull of the unit ball {f:q(f)≤1} of the quasi-normq, and letN be the null space of ρ. The normed envelope of WeakL 1, which we denote byW, is the space (WeakL 1/N, ρ). The Banach envelope of WeakL 1, $$\overline W $$ , is the completion ofW. We show that $$\overline W $$ is isometrically lattice isomorphic to a sublattice ofW. It is also shown that all rearrangement invariant Banach function spaces are isometrically lattice isomorphic to a sublattice ofW.