  We consider a non-Markovian discrete-time random walk on $\mathbb{Z}$ with
unbounded memory called the elephant random walk (ERW). We prove a strong
invariance principle for the ERW. More specifically, we prove that, under a
suitable scaling and in the diffusive regime as well as at the critical value
$p_c=3/4$ where the model is marginally superdiffusive, the ERW is almost
surely well approximated by a Brownian motion. As a by-product of our result we
get the law of iterated logarithm and the central limit theorem for the ERW.
