We introduce a ring resonator, which employs a one-dimensional phononic crystal on its inner surface, and investigate its performance as a gas sensor both numerically and experimentally. Having periodic equilateral trapezoidal protrusions, the ring resonator with 207 periods is optimized through band structure calculations via the finite-element method. A surface band linear around 58kHz is observed. The resonator exhibits sharp transmission peaks with a broad free-spectral range of 0.54kHz. Accordingly, a peak at 58.49kHz with a high-quality factor of 8196 appears. Application in detection of the carbon dioxide level in air with high sensitivity is demonstrated. The 58.49kHz peak red shifts linearly at 17.3mHz/ppm and 17.8mHz/ppm rates, as obtained from numerical calculations and experiments, respectively. Besides, the peak shape and maximum intensity are preserved. Due to the linear shift of the resonance peak with respect to the carbon dioxide concentration, acoustic intensity at initial peak frequency can be utilized as an auxiliary means for concentrations up to 1000ppm. The proposed ring resonator can be adapted to a variety of acoustic devices such as liquid concentration sensors based on phononic crystals, surface acoustic wave sensors, and micromechanical resonators.