Damped Ly{alpha} (DLA) and sub-DLA absorbers in quasar spectra provide the most sensitive tools for measuring the element abundances of distant galaxies. The estimation of abundances from absorption lines depends sensitively on the accuracy of the atomic data used. We have started a project to produce new atomic spectroscopic parameters for optical and UV spectral lines using state-of-the-art computer codes employing a very broad configuration interaction (CI) basis. Here we report our results for ZnII, an ion used widely in studies of the interstellar medium (ISM) as well as DLAs and sub-DLAs. We report new calculations of many energy levels of Zn II and the line strengths of the resulting radiative transitions. Our calculations use the CI approach within a numerical Hartree-Fock framework. We use both nonrelativistic and quasi-relativistic one-electron radial orbitals. We have incorporated the results of these atomic calculations into the plasma simulation code Cloudy and applied them to a lab plasma and examples of a DLA and a sub-DLA. Our values of the ZnII {lambda}{lambda}2026,2062 oscillator strengths are higher than previous values by 0.10dex. The Cloudy calculations for representative absorbers with the revised Zn atomic data imply ionization corrections lower than calculated earlier by 0.05dex. The new results imply that Zn metallicities should be lower by 0.1dex for DLAs and by 0.13-0.15dex for sub-DLAs than in past studies. Our results can be applied to other studies of ZnII in the Galactic and extragalactic ISM.