Let R be the ring of projective invariants of n labeled points in the projective plane under the action of SL₃(C). We give an upper bound, linear in n, on the degree in which R is generated. This result follows as a corollary of a more general result bounding the degrees in which type-A weight rings are generated. These rings arise as follows: Given a pair λ, μ of weights for SL₃(C) with λ dominant, let V_λ[μ] denote the μ-isotypic component of the irreducible representation V_λ with highest weight λ. A type-A weight ring R(λ, μ) is a graded ring with graded pieces V_{N λ}[N μ], where N ranges over the positive integers. Such a ring is the projective coordinate ring of a weight variety W(λ, μ). Using results of H. Derksen and V. L. Popov, we show that R(λ, μ) is generated in degree at most dimW(λ, μ).