The author has pioneered the study of completions of lattice ordered groups by means of Cauchy structures. This paper extends this study to arbitrary distributive lattices by means of Cauchy structures obtained from two intrinsic (and generally non-topological) lattice convergences called \(\alpha\) and \(\beta\). The \(\beta\)-convergence, which for infinitely distributive lattices coincides with Birkhoff's order convergence, leads to a Cauchy completion closely related to (but not always equal to) the MacNeille lattice completion. In the case of \(\alpha\)-convergence, repeated iteration of the Cauchy completion process may be required, but the end result is an ''essential'' extension which is an infinitely distributive, complete lattice.