With the rapid progress of optical sensors and optical communications, the laser source with low phase noise and intensity noise is in urgent demand. To meet the needs, a novel low-noise fiber laser with compound cavity structure is designed. The stable single-mode operation is realized by using a transient self-written fiber grating, produced by the effect of saturable absorption within a section of unpumped erbium-doped fiber, as narrow band filter for mode selecting and linewidth squeezing. Through optical path modification, the spectral signal-to-noise ratio is better than 62 dB. With opto-electronic negative feedback, the intensity noise of the source is well suppressed, the intensity noise in the low frequency band is greatly improved and the relaxation oscillation peak is reduced with about 25 dB. The laser has an output power more than 1 dBm, linewidth less than 1 kHz, and side-mode-suppression ratio larger than 50 dB. The excellent noise property indicates that the source has important applications in optical sensors and optical communications.