  We report the first measurement of the elliptic anisotropy ($v_2$) of the
charm meson $D^0$ at mid-rapidity ($|y|$\,$<$\,1) in Au+Au collisions at \sNN =
200\,GeV. The measurement was conducted by the STAR experiment at RHIC
utilizing a new high-resolution silicon tracker. The measured $D^0$ $v_2$ in
0--80\% centrality Au+Au collisions can be described by a viscous hydrodynamic
calculation for transverse momentum ($p_{\rm T}$) less than 4\,GeV/$c$. The
$D^0$ $v_2$ as a function of transverse kinetic energy ($m_{\rm T} - m_0$,
where $m_{\rm T} = \sqrt{p_{\rm T}^2 + m_0^2}$) is consistent with that of
light mesons in 10--40\% centrality Au+Au collisions. These results suggest
that charm quarks have achieved local thermal equilibrium with the medium
created in such collisions. Several theoretical models, with the
temperature--dependent, dimensionless charm spatial diffusion coefficient
($2{\pi}TD_s$) in the range of $\sim$2--12, are able to simultaneously
reproduce our $D^0$ $v_2$ result and our previously published results for the
$D^0$ nuclear modification factor.
