The production rates of D∗±D∗± mesons in charm and bottom events at centre-of-mass energies of about 91 GeV and the partial width of primary cc¯cc¯ pairs in hadronic Z0Z0 decays have been measured at LEP using almost 4.4 million hadronic Z0Z0 decays collected with the OPAL detector between 1990 and 1995. Using a combination of several charm quark tagging methods based on fully and partially reconstructed D∗±D∗± mesons, and a bottom tag based on identified muons and electrons, the hadronisation fractions of charm and bottom quarks into D∗±D∗± mesons have been found to be The fraction of cc¯cc¯ events in hadronic Z0Z0 decays, Γcc¯/Γhad=Γ(Z0→cc¯)/Γ(Z0→hadrons)Γcc¯/Γhad=Γ(Z0→cc¯)/Γ(Z0→hadrons) , is determined to be In all cases the first error is statistical, and the second one systematic. The last error quoted for Γcc¯/ΓhadΓcc¯/Γhad is due to external branching ratios.