  We study formation of stellar mass binary black holes (BBHs) originating from
Population III (PopIII) stars, performing stellar evolution simulations for
PopIII binaries with MESA. We find that a significant fraction of PopIII
binaries form massive BBHs through stable mass transfer between two stars in a
binary, without experiencing common envelope phases. We investigate necessary
conditions required for PopIII binaries to form BBHs coalescing within the
Hubble time with a semi-analytical model calibrated by the stellar evolution
simulations. The formation efficiency of coalescing PopIII BBHs is estimated
for two different initial conditions for PopIII binaries with large and small
separations, respectively. Consequently, in both models, $\sim 10\%$ of the
total PopIII binaries form BBHs only through stable mass transfer and $\sim
10\%$ of these BBHs merge due to gravitational wave emission within the Hubble
time. Furthermore, the chirp mass of merging BBHs has a flat distribution over
$15\lesssim M_{\rm chirp}/M_\odot \lesssim 35$. This formation pathway of
PopIII BBHs is presumably robust because stable mass transfer is less uncertain
than common envelope evolution, which is the main formation channel for
Population II BBHs. We also test the hypothesis that the BBH mergers detected
by LIGO originate from PopIII stars using our result and the total number of
PopIII stars formed in the early universe as inferred from the optical depth
measured by Planck. We conclude that the PopIII BBH formation scenario can
explain the mass-weighted merger rate of the LIGO's O1 events with the maximal
PopIII formation efficiency inferred from the Planck measurement, even without
BBHs formed by unstable mass transfer or common envelope phases.
