Haglund, Rhoades, and Shimozono recently introduced a quotient R_n,k of the polynomial ring Q[x₁, ..., x_n] depending on two positive integers k = n, which reduces to the classical coinvariant algebra of the symmetric group S_n if k = n. They determined the graded S_n-module structure of R_n,k and related it to the Delta Conjecture in the theory of Macdonald polynomials. We introduce an analogous quotient S_n,k and determine its structure as a graded module over the (type A) 0-Hecke algebra H_n(0), a deformation of the group algebra of S_n. When k = n we recover earlier results of the first author regarding the I>H_n(0)-action on the coinvariant algebra.