  The one-dimensional elephant random walk is a typical model of discrete-time
random walk with step-reinforcement, and is introduced by Sch\"{u}tz and
Trimper (2004). It has a parameter $\alpha \in (-1,1)$: The case $\alpha=0$
corresponds to the simple symmetric random walk, and when $\alpha>0$ (resp.
$\alpha<0$), the mean displacement of the walker at time $n$ grows (resp.
vanishes) like $n^{\alpha}$. The walk admits a phase transition at $\alpha=1/2$
from the diffusive behavior to the superdiffusive behavior. In this paper, we
study the rate of the moment convergence in the central limit theorem for the
position of the walker when $-1 < \alpha \leq 1/2$. We find a crossover
phenomenon in the rate of convergence of the $2m$-th moments with
$m=2,3,\ldots$ inside the diffusive regime $-1<\alpha<1/2$.
