In this work a narrow linewidth (1 kHz) laser source is used to measure the free spectral range of a fiberized Fabry-Perot etalon with sub-Hz accuracy (10(-8)). A previously demonstrated technique based on the Pound-Drever-Hall error signal is improved in accuracy by the use of a narrow linewidth laser swept in frequency via an acousto-optic modulator, or single sideband generation. The sub-Hz (10(-8)) accuracy attained enables the characterization of both the long-term drift and the polarization dependence of the free spectral range of the fiberized etalon.