  Drinfeld showed that any finite dimensional Hopf algebra \G extends to a
quasitriangular Hopf algebra \D(\G), the quantum double of \G. Based on the
construction of a so--called diagonal crossed product developed by the authors,
we generalize this result to the case of quasi--Hopf algebras \G. As for
ordinary Hopf algebras, as a vector space the ``quasi--quantum double'' \D(\G)
is isomorphic to the tensor product of \G and its dual \dG. We give explicit
formulas for the product, the coproduct, the R--matrix and the antipode on
\D(\G) and prove that they fulfill Drinfeld's axioms of a quasitriangular
quasi--Hopf algebra. In particular \D(\G) becomes an associative algebra
containing \G as a quasi--Hopf subalgebra. On the other hand, \dG \otimes 1 is
not a subalgebra of \D(\G) unless the coproduct on \G is strictly
coassociative. It is shown that the category of finite dimensional
representations of \D(\G) coincides with what has been called the double
category of \G--modules by S. Majid [M2]. Thus our construction gives a
concrete realization of Majid's abstract definition of quasi--quantum doubles
in terms of a Tannaka--Krein--like reconstruction procedure. The whole
construction is shown to generalize to weak quasi--Hopf algebras with \D(\G)
now being linearly isomorphic to a subspace of \dG \otimes \G.
