Kitchloo and Morava give a strikingly simple picture of elliptic cohomology at the Tate curve by studying a completed version of S1-equivariant K-theory for spaces. Several authors (cf. [2],[10],[11]) have suggested that an equivariant version ought to be related to the work of Freed-Hopkins-Teleman ([5],[6],[7]). However, a first attempt at this runs into apparent contradictions concerning twist, degree, and cup product. Several authors (cf. [3],[8],[9]) have solved the problem over the complex numbers by interpreting the S1-equivariant parameter as a complex variable and using holomorphicity as the technique for completion. This paper gives a solution that works integrally, by constructing a carefully completed model of K-theory for S1-equivariant stacks which allows for certain “convergent” infinite-dimensional cocycles.