Using the concept of weak measurements in classical optics, we introduce a new polarimetry tool that is capable of amplifying and quantifying extremely small sample polarization anisotropy effects, namely, linear and circular diattenuation and retardance. The approach is founded on interferometric realization of the weak value of polarization observable and is experimentally obtained by near destructive interference of polarized light in a Mach-Zehnder interferometer in the presence of a weak polarization anisotropy effect... Real and imaginary weak value amplification of a given sample anisotropy effect manifests in different Stokes vector elements at the exit port of the interferometer, which follows orthogonal trajectories in the Poincare sphere. Remarkably, the weak value polarimeter enhances the sensitivity of quantification of a small sample anisotropy effect exactly by the large weak value amplification factor using traditional Stokes vector measurements. This opens up a new paradigm in the domain of polarimetry by extending the measurement limit far beyond conventional polarimeters. (read more)