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 șs fine L2-norm estimates between the Wiener sausage and the Brownian intersection local times.