  A certain class of unitary representations of U_q(sl(2,R)) has the property
of being simultanenously a representation of U_{tilde{q}}(sl(2,R)) for a
particular choice of tilde{q}(q). Faddeev has proposed to unify the quantum
groups U_q(sl(2,R)) and U_{tilde{q}}(sl(2,R)) into some enlarged object for
which he has coined the name ``modular double''. We study the R-operator, the
co-product and the Haar-measure for the modular double of U_q(sl(2,R)) and
establish their main properties. In particular it is shown that the
Clebsch-Gordan maps constructed in [PT2] diagonalize this R-operator.
