We demonstrate the generation of 1.1 J pulses of picosecond duration at 1 kHz repetition rate (1.1 kW average power) from a diode-pumped chirped pulse amplification Yb:YAG laser. The laser employs cryogenically cooled amplifiers to generate λ=1030nm pulses with average power of up to 1.26 kW prior to compression with excellent beam quality. Pulses are compressed to 4.5 ps duration with 90% efficiency. This compact picosecond laser will enable a variety of applications that require high energy ultrashort pulses at kilohertz repetition rates.