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 $15lesssim M_{rm chirp}/M_odot lesssim 35$. 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 LIGOs 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.
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