In two recent papers, we constructed a new N→∞ limit of the W N algebras, which we denote W ∞ having generators of conformal spins 2, 3, …, with central terms for all spins. In this paper, we constrct another new algebra, which we denote W 1+∞, with generators of conformal spins, 1, 2, 3, …, again with central terms for all spins. The requirement that the algebras be closed requires that one include the spin-1 generators in W 1+∞, and prohibits their inclusion in W ∞. Paralleling our analogous construction for W ∞, we show that the new algebra can also be realised as the antisymmetric part of an associative “lone-star” product, which also closes on the set of generators with conformal spins ⩾1.