We have developed a 1-kHz repetition-rate Ti:sapphire laser system that can simultaneously generate high peak and average powers of 0.2TW and 4W, respectively. The laser system generates 4-mJ energy pulses with a 20-fs pulse width. We eliminated thermal lensing in the system by cooling the Ti:sapphire crystal to 125K. The output 20-fs pulses were fully characterized by use of the new technique of transient-grating frequency-resolved optical gating. We demonstrate experimentally that the pulse duration at the output is limited only by fifth-order dispersion.