The amplitude-phase coupling of a stochastically varying field has been predicted to lead to a shift of the effective resonance frequency when this field interacts with a two-level system. Experimentally, we have generated such a field by passing a semiconductor laser beam through a Michelson interferometer, thereby converting part of the intrinsic phase flucutations into amplitude fluctuations. We find indeed a shifted resonance frequency when this beam interacts with the Rb D& transition. Depending on the path-length diference of the interferometer, shifts of more than 10 MHz have been measured. The experiments have been performed in the weak-field limit. This allows a convolution-type interpretation of the stochastic realization shift. PACS number(s): 42.50.Ar, 42.25.Hz, 32.30.r, 07.65.b