Photon acceleration is the phenomenon whereby a light wave changes color when propagating through a medium whose index of refraction changes in time. This concept can be used to describe the spectral changes experienced by electromagnetic waves when they propagate in spatially and temporally varying plasmas. In this paper the detection of a large-amplitude laser-driven wake field is reported for the first time, demonstrating photon acceleration. Several features characteristic of photon acceleration in wake fields, such as splitting of the main spectral peak and asymmetries between the blueshift and redshift for large shifts, have been observed. The experiment is modeled using both a novel photon-kinetic code and a three-dimensional particle-in-cell code. In addition to the wide-ranging applications in the field of compact particle accelerators, the concept of wave kinetics can be applied to understanding phenomena in nonlinear optics, space physics, and fusion energy research.