We demonstrate an optical parametric chirped-pulse amplification (OPCPA) system with the pulse energy of 1.5 mJ at a 1 kHz repetition rate. The newly developed 100 ps Ti:sapphire pump laser system, which was optically synchronized with OPCPA seed pulses, delivered 10 mJ, 400 nm pump pulses. After three-stage parametric amplification, recompression of the amplifier output from 45 ps to 6.4 fs was performed. The pulse width of 6.4 fs is, to our knowledge, the shortest ever obtained by OPCPA, and the average power of 1.5 W (1.5 mJ, 1 kHz) is believed to be the highest among few-cycle OPCPA systems.