  The diffusion coefficient of Li$^+$ ions ($D_{\rm Li}$) in the battery
material Li$_x$CoO$_2$ has been investigated by muon-spin relaxation
($\mu^+$SR). Based on the experiments in zero-field and weak
longitudinal-fields at temperatures up to 400 K, we determined the fluctuation
rate ($\nu$) of the fields on the muons due to their interaction with the
nuclear moments. Combined with susceptibility data and electrostatic potential
calculations, clear Li$^+$ ion diffusion was detected above $\sim150$K. The
$D_{\rm Li}$ estimated from $\nu$ was in very good agreement with predictions
from first-principles calculations, and we present the $\mu^+$SR technique as
an % novel and optimal probe to detect $D_{\rm Li}$ for materials containing
magnetic ions.
