  In this paper, we carry out the ``quantum double construction'' of the
specific quantum groups we constructed earlier, namely, the ``quantum
Heisenberg group algebra'' (A,\Delta) and its dual, the ``quantum Heisenberg
group'' (\hat{A},\hat{\Delta}). Our approach will be by constructing a suitable
multiplicative unitary operator. In this way, we are able to retain the
C*-algebra framework, and thus able to carry out our construction within the
category of locally compact quantum groups. This construction is a kind of a
generalized crossed product. To establish that the quantum double we obtain is
indeed a locally compact quantum group, we will also discuss the dual of the
quantum double and the Haar weights on both of these double objects. Towards
the end, we also include a construction of a (quasitriangular) ``quantum
universal R-matrix''.
