The polarization of light conveys unique information that can be exploited by crucial applications. The bulky and costly discrete optical components used in conventional polarimeters limit their broad adoption. A compact, low-cost polarimeter would bring this functionality into a myriad of new scenarios and revolutionize its exploitation. Here we present a high-performance, full-Stokes polarimeter on a silicon chip. A surface polarization splitter and on-chip optical interferometer circuit produce the complete analysis matrix of an optimally conditioned polarimeter. A matrix analysis on measurement errors is also performed. This solid-state polarimeter is a system-on-a-chip with exceptional compactness, stability, and speed that could be used singly or in integrated arrays. Large arrays can increase the speed and resolution of full-Stokes imaging; therefore, our design provides a scalable polarimeter solution.