Stimulated Brillouin scattering is an effective way to achieve a laser with low background noise due to its narrowband gain. Based on a high Q fiber ring resonator, a Brillouin laser with low noise is studied. The pump light is locked into a single-mode ring resonator with a length of 8 m by the Pound-Drever-Hall (PDH) locking technique, and a backward Stokes light with a frequency difference of 10.81 GHz from the pump light is obtained. The frequency noise of the Stokes light is measured by the correlative delay self-heterodyne method. The experimental results show that the threshold of the Brillouin laser based on the fiber ring resonator is 5.3 mW. In the high-frequency part (the frequencyis greater than 10 kHz), the backward Stokes light suppresses the frequency noise of the pump light up to 30 dB, which is close to the theoretical suppression limit (34 dB).