  This paper is a continuation of "Quantization of Lie bialgebras, I"
(q-alg/9606005). We show that the quantization procedure defined in
"Quantization of Lie bialgebras, I" is given by universal acyclic formulas and
defines a functor from the category of Lie bialgebras to the category of
quantized universal enveloping algebras. We also show that this functor defines
an equivalence between the category of Lie bialgebras over k[[h]] and the
category quantized universal enveloping (QUE) algebras.
