  The programming new $e^{+}e^-$ collider with high luminosity shall provide
another useful platform to study the properties of the doubly heavy $B_c$ meson
in addition to the hadronic colliders as LHC and TEVATRON. Under the `New Trace
Amplitude Approach', we calculate the production of the spin-singlet $B_c$ and
the spin-triplet $B^*_c$ mesons through the $Z^0$ boson decays, where
uncertainties for the production are also discussed. Our results show
$\Gamma_{(^1S_0)}= 81.4^{+102.1}_{-40.5}$ KeV and
$\Gamma_{(^3S_1)}=116.4^{+163.9}_{-62.8}$ KeV, where the errors are caused by
varying $m_b$ and $m_c$ within their reasonable regions.
