Let F be a field. We show that certain subrings contained between the polynomial ring F[X] = F[X1, ⋯ ,Xn] and the power series ring F[X][[Y ]] = F[X1, ⋯ ,Xn][[Y ]] have Weierstrass Factorization, which allows us to deduce both unique factorization and the Noetherian property. These intermediate subrings are obtained from elements of F[X][[Y ]] by bounding their total X-degree above by a positive real-valued monotonic up function λ on their Y -degree. These rings arise naturally in studying the p-adic analytic variation of zeta functions over finite fields. Future research into this area may study more complicated subrings in which Y = (Y1, ⋯ , Ym) has more than one variable, and for which there are multiple degree functions, λ1, ⋯ , λm. Another direction of study would be to generalize these results to k-affinoid algebras. © 2010 American Mathematical Society.