We report the amplification of three-cycle, 8.5 fs optical pulses in a near-infrared noncollinear optical parametric chirped-pulse amplifier (OPCPA) up to energies of 80 mJ. Improved dispersion management in the amplifier by means of a combination of reflection grisms and a chirped-mirror stretcher allowed us to recompress the amplified pulses to within 6% of their Fourier limit. The novel ultrabroad, ultraprecise dispersion control technology presented in this work opens the way to scaling multiterawatt technology to even shorter pulses by optimizing the OPCPA bandwidth.