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Considering the possibility to build an $e^+ e^-$ collider at the energiesaround the $Z^0$-boson resonance with a planned luminosity so high as ${\calL}\propto 10^{34}\sim 10^{36}cm^{-2}s^{-1}$ (super $Z$-factory), we make adetailed discussion on the $(c\bar{b})$-quarkonium production through$e^{+}+e^{-}\rightarrow (c\bar{b})[n]+b+\bar{c}$ within the framework ofnon-relativistic QCD. To simplify the hard-scattering amplitude as much aspossible and to derive analytic expressions for the purpose of future eventssimulation, we adopt the "improved trace technology" to do our calculation,which deals with the hard scattering amplitude directly at the amplitude levelother than the conventional way at the squared-amplitude level. Totalcross-section uncertainties caused by the quark masses are predicted by taking$m_c=1.50\pm0.30$ GeV and $m_b=4.90\pm0.40$ GeV. If all higher$(c\bar{b})$-quarkonium states decay to the ground state $B_c$($|(c\bar{b})_{\bf 1}[^1S_0]>$) with 100% efficiency, we obtain$\sigma_{e^{+}+e^{-}\rightarrow B_{c}+b+\bar{c}} =5.190^{+6.222}_{-2.419}$ pb,which shows that about $10^5 \sim 10^7$ $B_c$ events per operation year can beaccumulated in the super $Z$-factory. If taking the collider energy runsslightly off the $Z^0$-peak, i.e. $\sqrt{S}=(1.00\pm0.05) m_Z$, the totalcross-section shall be lowered by about one-order from its peak value. Such asuper $Z$-factory shall provide another useful platform to study the propertiesof $B_c$ meson, or even the properties of its excited $P$-wave states, inaddition to its production at the hadronic colliders Tevatron and LHC.

$B_c$ Meson Production around the $Z^0$ Peak at a High Luminosity $e^+ e^-$ Collider