Let 𝔤 be a complex Lie algebra, and let U(𝔤) be its universal enveloping algebra. We study homological properties of topological Hopf algebras containing U(𝔤) as a dense subalgebra. Specifically, let θ: U(𝔤) → H be a homomorphism to a topological Hopf algebra H. Assuming that H is either a nuclear Fréchet space or a nuclear (DF)-space, we formulate conditions on the dual algebra, H', that are sufficient for H to be stably flat over U(𝔤) in the sense of A. Neeman and A. Ranicki (2001) (i.e., for θ to be a localization in the sense of J. L. Taylor (1972)). As an application, we prove that the Arens-Michael envelope, Ũ(𝔤), of U(𝔤) is stably flat over U(𝔤) provided 𝔤 admits a positive grading. We also show that R. Goodman's (1979) weighted completions of U(𝔤) are stably flat over U(𝔤) for each nilpotent Lie algebra 𝔤, and that P. K. Rashevskii's (1966) hyperenveloping algebra is stably flat over U(𝔤) for any 𝔤. Finally, the algebra 𝓐(G) of analytic functionals (introduced by G. L. Litvinov (1969)) on the corresponding connected, simply connected complex Lie group G is shown to be stably flat over U(𝔤) precisely when 𝔤 is solvable.