Experimentally measured resonance hyper-Raman (RHR) spectra spanning the S(1) <- S(0), S(2) <- S(0), and S3 <- S(0) transitions in rhodamine 6G (R6G) have been recorded. These spectra are compared to the results of first-principles calculations of the RHR intensity that include both Franck Condon (A-term) and non-Condon (B-term) scattering effects. Good agreement between the experimental and theoretical results is observed, demonstrating that first-principles calculations of hyper-Raman intensities are now possible for large molecules such as R6G. Such agreement indicates that RHR spectroscopy will now be a routine aid for probing multiphoton processes. This work further shows that optimization of molecular properties to enhance either A- or B-term scattering might yield molecules with significantly enhanced two-photon properties.