We present an integrated design to sensitively measure changes in optical frequency using weak value amplification with a multi-mode interferometer. The technique involves introducing a weak perturbation to the system and then post-selecting the data in such a way that the signal is amplified without amplifying the technical noise, as has previously been demonstrated in a free-space setup. We demonstrate the advantages of a Bragg grating with two band gaps for obtaining simultaneous, stable high transmission and high dispersion. The device is more robust and easily scalable than the free-space implementation, and provides amplified sensitivity compared to other methods of measuring changes in optical frequency on a chip, such as an integrated Mach-Zehnder interferometer.