  We prove an analogue of the Poincare'-Birkhoff-Witt theorem and of the
Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive
part of a cofree Hopf algebra is a B-infini-algebra. We construct a universal
enveloping functor U2 from B-infini-algebras to 2-associative algebras, i.e.
algebras equipped with two associative operations. We show that any cofree Hopf
algebra H is of the form U2(Prim H). We take advantage of the simple
description of the free 2as-algebra in terms of planar trees to unravel the
structure of the operad B-infini.
