  In this paper we prove that the centered three-dimensional Wiener sausage can
be strongly approximated by a one-dimensional Brownian motion running at a
suitable time clock. The strong approximation gives all possible laws of
iterated logarithm as well as the convergence in law in terms of process for
the normalized Wiener sausage. The proof relies on Le Gall's estimates between
the Wiener sausage and the Brownian intersection local times.
