A bound-to-free transition initiated by femtosecond excitation of diatomic molecules results in photofragments with a distribution of kinetic energies. A measurement of the kinetic-energy distribution yields the modulus squared of the asymptotic momentum-space wave packet prepared in the laser excitation process. On the other hand, the coordinate-space density of the wave packet entering the interaction-free region can be determined from pump-probe integrated fluorescence spectroscopy. We provide several numerical examples to show that this information can be used to determine the phase of the asymptotic wave packet so that this particular quantum-mechanical wave function can be characterized completely. To achieve this aim we use an iteration scheme (Gerchberg-Saxton algorithm) which does not require any further information about the system or the laser pulses.