Intense Compton-scattering γ-ray radiation driven by laser wakefield acceleration (LWFA) and generation of ultrashort positron beams are investigated by Monte Carlo simulation. Using an LWFA driven GeV electron bunch and a 45 femtosecond, 90 mJ/pulse, and 10 Hz Ti:Sapphire laser for driving the Compton scattering, fs γ-ray pulses were generated. The latter have a flux of ≥108/s, peak brightness of ≥1020 photons/(s mm2 mrad2 0.1% bandwidth), and photon energy of 5.9 to 23.2 MeV. The γ-ray pulses then impinge on a thin high-Z target. More than 107 positrons/s in the form of sub-100 fs pulses at several MeV can be produced. Such ultrashort positron pulses can be useful as the pump-probe type positron annihilation spectroscopy as well as in other applications.