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We present the merger rate density of Population (Pop.) III binary blackholes (BHs) by means of a widely-used binary population synthesis code BSE withextensions to very massive and extreme metal-poor stars. We consider not onlylow-mass BHs (lBHs: $5-50 M_\odot$) but also high-mass BHs (hBHs: $130-200M_\odot$), where lBHs and hBHs are below and above the pair-instability massgap ($50-130 M_\odot$), respectively. Pop. III BH-BHs can be categorized intothree subpopulations: BH-BHs without hBHs (hBH0s: $m_{\rm tot} \lesssim 100M_\odot$), with one hBH (hBH1s: $m_{\rm tot} \sim 130-260 M_\odot$), and withtwo hBHs (hBH2s: $m_{\rm tot} \sim 270-400 M_\odot$), where $m_{\rm tot}$ isthe total mass of a BH-BH. Their merger rate densities at the current universeare $\sim 0.1$ yr$^{-1}$ Gpc$^{-3}$ for hBH0s, and $\sim 0.01$ yr$^{-1}$Gpc$^{-3}$ for the sum of hBH1s and hBH2s, provided that the mass density ofPop. III stars is $\sim 10^{13} M_\odot$ Gpc$^{-3}$. These rates are modestlyinsensitive to initial conditions and single star models. The hBH1 and hBH2mergers can dominate BH-BHs with hBHs discovered in near future. They have loweffective spins $\lesssim 0.2$ in the current universe. The number ratio of thehBH2s to the hBH1s is high, $\gtrsim 0.1$. We also find BHs in the mass gap (upto $\sim 85 M_\odot$) merge. These merger rates can be reduced to nearly zeroif Pop. III binaries are always wide ($\gtrsim 100 R_\odot$), and if Pop. IIIstars always enter into chemically homogeneous evolution. The presence of closePop. III binaries ($\sim 10 R_\odot$) are crucial for avoiding the worstscenario.

Merger rate density of Population III binary black holes below, above, and in the pair-instability mass gap