The Artin-Tate lemma states that if $A \subseteq B \subseteq C$ are commutative rings where $A$ and $C$ are Noetherian, $C$ is finitely generated as an $A$-algebra, and $C$ is finitely generated as a $B$-module, then $B$ is finitely-generated as an $A$-algebra.
I am wondering about a kind of "complete local" analogue of the above. Specifically, let $(C,\mathfrak m)$ be a complete Noetherian local ring, let $A$ be a coefficient ring (as in the Cohen structure theorem) [or perhaps just a complete Noetherian local subring with topology compatible with that of $C$?], and let $B$ be ring between $A$ and $C$ such that $C$ is finite as a $B$-module. Does it follow that $B$ is $(\mathfrak m \cap B)$-adically complete?
(Recall that Eakin's theorem guarantees that $B$ is Noetherian.)