It is shown both theoretically and experimentally that the phase of a transition dipole matrix element may be determined from the interference between competing quantum mechanical paths. Specifically, we consider simultaneous m- and n-photon excitation of a molecule to a degenerate continuum state, leading to multiple reaction products. Variation of the relative phase of the electromagnetic fields produces a modulation of the product signals in the various channels. A relation between the phase lag between a pair of product signals and properties of the continuum is established. Information is revealed about coupling mechanisms, the properties of weak spectroscopic transitions, and the phases of quasi-bound wave functions.