  We present the merger rate density of Population (Pop.) III binary black
holes (BHs) by means of a widely-used binary population synthesis code BSE with
extensions to very massive and extreme metal-poor stars. We consider not only
low-mass BHs (lBHs: $5-50 M_\odot$) but also high-mass BHs (hBHs: $130-200
M_\odot$), where lBHs and hBHs are below and above the pair-instability mass
gap ($50-130 M_\odot$), respectively. Pop. III BH-BHs can be categorized into
three subpopulations: BH-BHs without hBHs (hBH0s: $m_{\rm tot} \lesssim 100
M_\odot$), with one hBH (hBH1s: $m_{\rm tot} \sim 130-260 M_\odot$), and with
two hBHs (hBH2s: $m_{\rm tot} \sim 270-400 M_\odot$), where $m_{\rm tot}$ is
the total mass of a BH-BH. Their merger rate densities at the current universe
are $\sim 0.1$ 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 of
Pop. III stars is $\sim 10^{13} M_\odot$ Gpc$^{-3}$. These rates are modestly
insensitive to initial conditions and single star models. The hBH1 and hBH2
mergers can dominate BH-BHs with hBHs discovered in near future. They have low
effective spins $\lesssim 0.2$ in the current universe. The number ratio of the
hBH2s to the hBH1s is high, $\gtrsim 0.1$. We also find BHs in the mass gap (up
to $\sim 85 M_\odot$) merge. These merger rates can be reduced to nearly zero
if Pop. III binaries are always wide ($\gtrsim 100 R_\odot$), and if Pop. III
stars always enter into chemically homogeneous evolution. The presence of close
Pop. III binaries ($\sim 10 R_\odot$) are crucial for avoiding the worst
scenario.
