  For a flat commutative $k$-algebra $A$ such that the enveloping algebra
$A\otimes_k A$ is noetherian, given a finitely generated bimodule $M$, we show
that the adic completion of the Hochschild cohomology module $HH^n(A/k,M)$ is
naturally isomorphic to $HH^n(\widehat{A}/k,\widehat{M})$. To show this, we (1)
make a detailed study of derived completion as a functor $D(A) \to
D(\widehat{A})$ over a non-noetherian ring $A$; (2) prove a flat base change
result for weakly proregular ideals; and (3) Prove that Hochschild cohomology
and analytic Hochschild cohomology of complete noetherian local rings are
isomorphic, answering a question of Buchweitz and Flenner. Our results makes it
possible for the first time to compute the Hochschild cohomology of
$k[[t_1,\dots,t_n]]$ over any noetherian ring $k$, and open the door for a
theory of Hochschild cohomology over formal schemes.
