Recently, it has been shown that a slow-light augmented unbalanced Mach-Zehnder interferometer (SLAUMZI) can be used to enhance significantly the sensitivity of measuring the frequency shift of a laser, compared to the heterodyne technique. Here, we show that a similar enhancement can be realized using a slow-light augmented Fabry-Perot Cavity (SLAFPC), due to the fact that an FPC is inherently unbalanced, since different bounces of the field traverse different path lengths before interfering with the other bounces. We show how the degree of enhancement in sensitivity depends on the spectral width of the laser and the finesse of the FPC. For potentially realizable conditions, we show that a sensitivity enhancement factor ~2.8*10^6 can be achieved using a SLAFPC.