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 ^{3}He(n,p)^{3}H, using the capture of polarized cold neutrons in an unpolarized active ^{3}He 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 A_{PV}=[1.55±0.97(stat)±0.24(sys)]×10^{-8}, which has the smallest uncertainty of any hadronic parity-violating asymmetry measurement so far.