A stable and broadband microwave photonic phase shifter based on the combined use of a linear chirped fiber Bragg grating and optical single-sideband (OSSB) modulation is proposed and experimentally demonstrated. The quality of the radio frequency (RF) signal is improved by the spectral separation delay processing. The theoretical fundamentals of the scheme are explained and the phase shift can be controlled linearly by the wavelength of the light source. In the experiment, a full 360° phase shift with a 10 GHz bandwidth can be achieved and tuned dynamically, continuously, and stably.