We have built and used on several occasions an optical broadband stellar polarimeter, PlanetPol, which employs photoelastic modulators and avalanche photodiodes and achieves a photon-noise-limited sensitivity of at least 1 in 10⁶ in fractional polarization. Observations of a number of polarized standards taken from the literature show that the accuracy of polarization measurements is ~1%. We have developed a method for accurately measuring the polarization of altitude-azimuth mounted telescopes by observing bright nearby stars at different parallactic angles, and we find that the on-axis polarization of the William Herschel Telescope is typically ~15 × 10⁻⁶, measured with an accuracy of a few parts in 10⁷. The nearby stars (distance less than 32 pc) are found to have very low polarizations, typically a few ×10⁻⁶, indicating that very little interstellar polarization is produced close to the Sun and that their intrinsic polarization is also low. Although the polarimeter can be used for a wide range of astronomy, the very high sensitivity was set by the goal of detecting the polarization signature of unresolved extrasolar planets.