We present an experimental technique that permits mechanical-noise-free, cavity-enhanced frequency measurements of an atomic transition and its hyperfine structure. We employ the 532-nm frequency-doubled output from a Nd:YAG laser and an iodine vapor cell. The cell is placed in a folded ring cavity (FRC) with counterpropagating pump and probe beams. The FRC is locked with the Pound-Drever-Hall technique. Mechanical noise is rejected by differencing the pump and probe signals. In addition, this differenced error signal provides a sensitive measure of differential nonlinearity within the FRC.