The differential cross sections of the Σ^{-}p→Λn reaction were measured accurately for the Σ^{-} momentum (p_{Σ}) ranging from 470 to 650 MeV/c at the J-PARC Hadron Experimental Facility. Precise angular information about the Σ^{-}p→Λn reaction was obtained for the first time by detecting approximately 100 reaction events at each angular step of Δcosθ=0.1. The obtained differential cross sections show a slightly forward-peaking structure in the measured momentum regions. The cross sections integrated for -0.7≤cosθ≤1.0 were obtained as 22.5±0.68 [statistical error(stat.)] ±0.65 [systematic error(syst.)] mb and 15.8±0.83(stat)±0.52(syst) mb for 470<p_{Σ}(MeV/c)<550 and 550<p_{Σ}(MeV/c)<650, respectively. These results show a drastic improvement compared with past measurements of the hyperon-proton scattering experiments. They will play essential roles in updating the theoretical models of the baryon-baryon interactions.