In this paper we present a new proof of the homological stability of the moduli space of closed surfaces in a simply connected background space K , which we denote by S g (K) . The homology stability of surfaces in K with an arbitrary number of boundary components, S g,n (K) , was studied by the authors in a previous paper. The study there relied on stability results for the homology of mapping class groups, Γ g,n with certain families of twisted coefficients. It turns out that these mapping class groups only have homological stability when n , the number of boundary components, is positive, or in the closed case when the coefficient modules are trivial. Because of this we present a new proof of the rational homological stability for S g (K) , that is homotopy theoretic in nature. We also take the opportunity to prove a new stability theorem for closed surfaces in K that have marked points.