[en] The differential cross-section for K^-p backward elastic scattering was measured in steps of 10-20 MeV/c between 476 and 1084 MeV/c at the Brookhaven AGS. A negative kaon beam was focused onto a 21 cm long liquid hydrogen target and the forward scattered protons were identified kinematically by means of a single arm magnetic spectrometer. A Monte Carlo computer program aided in the subtraction of background events and in the calculation of correction factors. The new data have an average statistical precision of 2.5% which is an order of magnitude improvement over previous results. Together with a new partial wave analysis, the new data give additional evidence for a resonance between 1700 and 1725 MeV. The best fit to the data finds this to be a D₁₃ resonance with a mass of 1708 MeV and a width of 27 MeV. Comparable improvement in precision was also obtained in a measurement of the differential cross section for K^-p → Σ^-π⁺ in which the π⁺ emerges at 0⁰. 47 references