  Rota-Baxter associative algebras and Rota-Baxter Lie algebras are both
important in mathematics and mathematical physics, with the former a basic
structure in quantum field renormalization and the latter a operator form of
the classical Yang-Baxter equation. An outstanding problem posed by Gubarev is
to determine whether there is a Poincar\'e-Birkhoff-Witt theorem for the
universal enveloping Rota-Baxter associative algebra of a Rota-Baxter Lie
algebra. This paper resolves this problem positively, working with operated
algebras and applying the method of Gr\"obner-Shirshov bases.
