We have developed a software-based polarization spectrometer, PolariS, to acquire full-Stokes spectra with a very high spectral resolution of 61 Hz. The primary aim of PolariS is to measure the magnetic fields in dense star-forming cores by detecting the Zeeman splitting of molecular emission lines. The spectrometer consists of a commercially available digital sampler and a Linux computer. The computer is equipped with a graphics processing unit (GPU) to process FFT and cross-correlation using the Compute Unified Device Architecture (CUDA) library developed by NVIDIA. Thanks to a high degree of precision in quantization of the analog-to-digital converter and arithmetic in the GPU, PolariS offers excellent performances in linearity, dynamic range, sensitivity, bandpass flatness and stability. The software has been released under the MIT License and is available to the public. In this paper, we report the design of PolariS and its performance verified through engineering tests and commissioning observations.