The authors prove that finitely generated Coxeter groups act properly discontinuously on a locally finite, finite-dimensional CAT\((0)\) cube complex. Part of the motivation for this result was an attempt to show that Coxeter groups are biautomatic. In order to conclude this using a result of the authors' [Geom. Topol. 1, 1-7 (1997; Zbl 0887.20016)] the group must act properly discontinuously and cocompactly on a CAT\((0)\) cube complex. In general, the action constructed in this paper is not cocompact. The authors explore the failure of the action to be cocompact using a characterization of \textit{B. T. Williams} [Two topics in geometric group theory. Ph.D. thesis. Univ. Southampton (1999)] and are able to conclude that the action is cocompact in the case of right-angled Coxeter groups and word-hyperbolic Coxeter groups. Thus, they provide a new proof that these classes of groups are biautomatic.