It is shown here that Brillouin amplification can be used to produce picosecond pulses of petawatt power. Brillouin amplification is far more resilient to fluctuations in the laser and plasma parameters than Raman amplification, making it an attractive alternative to Raman amplification. Through analytic theory and multi-dimensional computer simulations, a novel, well-defined parameter regime has been found, distinct from that of Raman amplification, where pump-to-probe compression ratios of up to 100 and peak laser fluences over 1 kJ/cm$^2$ with 30\% efficiency have been achieved. High pulse quality has been maintained through control of parasitic instabilities.