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Representation theory for the Jordanian quantum algebra Uh(sl(2)) is developed using a nonlinear relation between its generators and those of sl(2). Closed form expressions are given for the action of the generators of Uh(sl(2)) on the basis vectors of finite dimensional irreducible representations. In the tensor product of two such representations, a new basis is constructed on which the generators of Uh(sl(2)) have a simple action. Using this basis, a general formula is obtained for the Clebsch-Gordan coefficients of Uh(sl(2)). Some remarkable properties of these Clebsch-Gordan coefficients are derived.