New emission branching fraction measurements for 898 lines of the first spectrum of cobalt (Co I) are determined from hollow cathode lamp spectra recorded with the National Solar Observatory 1m Fourier transform spectrometer on Kitt Peak, AZ and a high-resolution echelle spectrometer. Published radiative lifetimes from laser induced fluorescence measurements are combined with the branching fractions to determine accurate absolute atomic transition probabilities for the 898 lines. Hyperfine structure (hfs) constants for levels of neutral Co in the literature are surveyed and selected values are used to generate complete hfs component patterns for 195 transitions of Co I. These new laboratory data are applied to determine the Co abundance in the Sun and metal-poor star HD 84937, yielding log{epsilon}(Co)=4.955+/-0.007 ({sigma}=0.059) based on 82 Co I lines and log{epsilon}(Co)=2.785+/-0.008 ({sigma}=0.065) based on 66 Co I lines, respectively. A Saha or ionization balance test on the photosphere of HD 84937 is performed using 16 UV lines of Co II, and good agreement is found with the Co I result in this metal-poor ([FeI/H]=-2.32, [FeII/H]=-2.32) dwarf star. The resulting value of [Co/Fe]=+0.14 supports a rise of Co/Fe at low metallicity that has been suggested in other studies.