In this paper, we introduce the notion of σ−ideally Connes amenable for dual Banach algebras and give some hereditary properties for this new notion. We also investigate σ−ideally Connes amenability of ℓ1(G,ω). We show that if ω is a diagonally bounded weight function on discrete group G and σ is isometrically isomorphism of ℓ1(G,ω), then ℓ1(G,ω) is σ−ideally Connes amenable and so it is ideally Connes amenable.