We introduce the strict-weak polymer model, and show the KPZ universality of the free energy fluctuation of this model for a certain range of parameters. Our proof relies on the observation that the discrete time geometric q-TASEP model, studied earlier by Borodin and Corwin, scales to this polymer model in the limit q→1. This allows us to exploit the exact results for geometric q-TASEP to derive a Fredholm determinant formula for the strict-weak polymer, and in turn perform rigorous asymptotic analysis to show KPZ scaling and GUE Tracy–Widom limit for the free energy fluctuations. We also derive moments formulae for the polymer partition function directly by Bethe ansatz, and identify the limit of the free energy using a stationary version of the polymer model.National Science Foundation (U.S.) (grant DMS-1208998)Microsoft Research (Schramm Memorial Fellowship)Clay Mathematics Institute (Clay Research Fellowship)Institut Henri Poincare (Poincaré Chair)