  We report the first measurement of rapidity-odd directed flow ($v_{1}$) for
$D^{0}$ and $\overline{D^{0}}$ mesons at mid-rapidity ($|y| < 0.8$) in Au+Au
collisions at $\sqrt{s_{\rm NN}}$ = 200\,GeV using the STAR detector at the
Relativistic Heavy Ion Collider. In 10--80\% Au+Au collisions, the slope of the
$v_{1}$ rapidity dependence ($dv_{1}/dy$), averaged over $D^{0}$ and
$\overline{D^{0}}$ mesons, is -0.080 $\pm$ 0.017 (stat.) $\pm$ 0.016 (syst.)
for transverse momentum $p_{\rm T}$ above 1.5~GeV/$c$. The absolute value of
$D^0$-meson $dv_1/dy$ is about 25 times larger than that for charged kaons,
with 3.4$\sigma$ significance. These data give a unique insight into the
initial tilt of the produced matter, and offer constraints on the geometric and
transport parameters of the hot QCD medium created in relativistic heavy-ion
collisions.
