A low-noise single-frequency laser is a key component of the space-based gravitational wave detector, and the intensity noise of the laser directly affect the sensitivity of the space-based GW detector. In this work, we report a low-noise single-frequency laser designed for space-based gravitational wave detector. The laser is based on a master oscillator power amplifier (MOPA) design with a low-power, narrow-linewidth seed laser as MO and an all polarization-maintaining fiber amplifier as PA. The amplifier uses a robust mechanical design consists of an Yb-doped double-clad fiber forward pumped by wavelength-locked 976 nm pump LD to achieve 2.13 W of output power and 70 dB of SNR. To suppress the relative intensity noise (RIN) in the milli-Hertz regime (1 mHz -1 Hz), we characterize the power stabilization of a pump diode laser based on a PID feedback control loop using an in-loop photodetector. The power fluctuation can be converted into the fluctuation of the current signal by the photodiode, the current signal is converted into the voltage signal and amplified by the transimpedance circuit. Then, the voltage signal is compared with the voltage reference signal, and the error signal is achieved to adjust real-timely the drive current of the pump laser diode. This is a good way to significantly suppress the RIN of a laser at low frequency, but the measured RIN below 4 mHz is still higher than -60 dBc/Hz. In order to further suppress the RIN below 4 mHz, active precise temperature control technology is used to suppress the thermal noise of pump LD and fiber coupler. To assess the RIN milli-Hertz regime, we design a RIN measurement system consisting of a high-precision signal acquisition card (24 bit) and a computer program based on LabVIEW. The measurement range of the system is 2 μHz-102.4 kHz and the frequency resolution up to 2 µHz, much better than commercial instruments. By stabilizing the fiber amplifier pump LD current and the temperature of pump LD and fiber coupler, the measured out-of-loop RINs of -63.4 dBc/Hz@1 mHz and -105.8 dBc/Hz@1 Hz are achieved, and in the milli-Hertz regime of 1mHz-1Hz, the RIN is below -60 dBc/Hz. The results show that the feedback control of the fiber amplifier pump LD current and the temperature control of the key devices can effectively suppress the RIN in the millihertz frequency band, which lays a foundation for further improving the intensity noise performance in the low frequency band.