Optical-based rotation sensors have revolutionized precision, high-sensitivity inertial navigation systems. At the same time these sensors use bulky optical fiber spools or free-space resonators. A chip-based, micro-optical gyroscope is demonstrated that uses counterpropagating Brillouin lasers to measure rotation as a Sagnac-induced frequency shift. Preliminary work has demonstrated a rotation-rate measurement that surpasses prior micro-optical rotation-sensing systems by over 40-fold.