Polarized gamma-ray beams are a precise and selective probe for studying fundamental questions about nuclear structure and hadron properties. Improvements to this probe require new experimental approaches that can produce high-flux gamma-ray beams with easily switchable pure polarization states. Here, we report an optics-free method to precisely control the polarization of a Compton gamma-ray beam. Using a free-electron laser (FEL) oscillator with two helical undulator magnets of opposite helicities, we have produced a linearly polarized FEL beam with a variable polarization direction and an unprecedented degree of linear polarization, PLin = 0.997. With this FEL as a photon drive, we are able to generate Compton gamma-ray beams having either left/right-circular polarization or rotatable linear polarization. The linearly polarized gamma-ray beam has been characterized and shows PLin = 0.97. This demonstrated polarization control technique is well suited for high-flux gamma-ray production with any level of FEL power.