The programming new e + e − collider with high luminosity shall provide another useful platform to study the properties of the doubly heavy Bc 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 Bc and the spin-triplet B * c mesons through the Z 0 boson decays, where uncertainties for the production are also discussed. Our results show Γ ( PACS numbers: 12.38. Bx, 12.39.Jh, 14.40Lb, 14.40.Nd The B c meson is a double heavy quark-antiquark bound state and carries flavors explicitly. Since its first discovery at TEVATRON [1], B c physics is attracting more and wide interests. Recently, many progresses have been made for the hadronic production of B c meson at high energy colliders as LHC and TEVATRON. A computer program BCVEGPY for the direct hadronic production of B c meson has been presented in Refs. [2,3]. And it has been found that the indirect production of B c via top quark decays can also provide useful information on B c meson [4][5][6][7].Comparing with the hadronic colliders, an e + e − collider has its own advantages, mainly because of its lower background. As for the previous LEP-I experiment, no B c events have been found due to its lower collision energy and low luminosity [8,9]. However, if the luminosity of the e + e − collider can be raised up to L ∝ 10 34 cm −2 s −1 or even higher as programmed by the Internal Linear Collider (ILC) [10], then there might have enough events. Moreover, if the e + e − collider further runs at the Z 0 -boson energy, the resonance effects at the Z 0 peak may raise the production rate up to several orders. It has been estimated by Ref.[11] that more than 10 9∼10 Z 0 -events can be produced at ILC per year, which is about 3 ∼ 4 orders higher than that collected by LEP-I. Such a high luminosity collider is called as GigaZ [11] or a Z-factory [12]. Then it will open new opportunities not only for high precision physics in the electro-weak sector, but also for the hadron physics.The production of B c through Z 0 decays has been studied in Refs [8,9,13] with various methods. Since the process is very complicated, it would be helpful to have a cross check of these results. Furthermore, considering the forthcoming Z-factory, it may be interesting to know the theoretical uncertainties in estimating of B c production.For the purpose, we need to calculate the processc +b+c, whose Feynman diagrams are shown in Fig.(1). According to the NRQCD factorization formula * e-mail:wuxg@cqu.edu.cnc[14], the decay width for the process Z 0 → B ( * ) c + b +c can be written in the following factorization form:where the matrix element O H (n) is proportional to the inclusive transition probability of the perturbative state cb[n] into the bound states of B c . As for the two colorsinglet S-wave states cb[are related with the Bethe-Salpeter wave function at the origin that can be determined by the potential model [15][16][17][18][19][20]. dΓ(Z 0 → cb[n] + b +c) stands for the short-distan...