We report the first precision measurement of the parity-violating asymmetry in the direction of proton momentum with respect to the neutron spin, in the reaction 3He(n,p)3H, using the capture of polarized cold neutrons in an unpolarized active 3He target. The asymmetry is a result of the weak interaction between nucleons, which remains one of the least well-understood aspects of electroweak theory. The measurement provides an important benchmark for modern effective field theory and potential model calculations. Measurements like this are necessary to determine the spin-isospin structure of the hadronic weak interaction. Our asymmetry result is APV=[1.55±0.97(stat)±0.24(sys)]×10−8, which has the smallest uncertainty of any hadronic parity-violating asymmetry measurement so far.