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Measuring a synchronous phase shift as a function of beam current is commonly done in accelerator physics to estimate the longitudinal impedance of a storage ring vacuum chamber. This measurement is normally done with RF techniques that do not have enough accuracy to detect small phase shifts typical to the newer storage rings. In this paper we report results from a new method for precise measurement of a synchronous phase shift. Our method involves downmixing from the RF frequency to a kHz range and then using an audio DSP lock-in amplifier for the actual phase detection. This paper describes the idea and the advantages of a new method as well as its practical implementation in the apparatus we build for precise synchronous phase measurements in the Stanford Linear Collider damping rings. The results of those measurements are also presented

New apparatus for precise synchronous phase shift measurements in storage rings