We have developed a low-noise microwave frequency synthesizer for the world's first ⁸⁵Rb fountain clock. The setup is based on the techniques of low phase-noise frequency synthesis and phase-locking, which can realize the clock frequency signal output with low phase noise. When the output frequency is about 3.035 GHz and the frequency resolution is 0.7 μHz, the single-sideband phase noise densities at 1 Hz, 100 Hz, and 10 kHz Fourier frequency are -97 dBc·Hz ^-1, -127 dBc·Hz ^-1, and -130 dBc·Hz ^-1 , respectively. Moreover, the residual phase noise of the microwave frequency synthesizer is lower by 20 dB than that of the local oscillator. By comparing the contributions of quantum project noise and Dick effect, we show that the stability of the ⁸⁵Rb fountain clock is comparable to those of ⁸⁷Rb clock and ¹³³Cs clock, where the noise from a microwave system is the main noise source. The stability caused by the microwave source for the ⁸⁵Rb fountain clock is estimated as 2.9×10 ^-13τ^-1/2(τ is integration time), among which the contribution of the residual phase noise from the frequency synthesizer is 1.2×10 ^-14τ^-1/2. The synthesizer is helpful to realize the operation of the ⁸⁵Rb fountain clock with high stability and is expected to be the basis of the future improvement in clock performances.