We have investigated the RbCs 640 run system by mass-resolved resonance enhanced two-photon ionization in a cold molecular beam. Very complex vibronic structures were observed between 15420 and 15990 cm(-1). The parallel transitions of 2 (3)Pi(0) nu' = 4-20 <- X (1)Sigma(+) nu '' = 0 were identified by rotationally resolved spectra. Molecular constants and a Rydberg-Klein-Rees potential energy curve of the 2 (3)Pi(0) state were determined. The regular vibrational spacing of the parallel transition indicated that the 2 (3)Pi(0) state is not significantly perturbed by nearby excited electronic states. The complexity of the observed vibronic structures has been attributed to the coupled perpendicular transitions of 2 (1)Pi , 2 (3)Pi(1) , and 3 Sigma (3)Sigma(+)(1)nu '' = 0. For the perpendicular bands observed in the lower-energy spectral region between 15420 and 15630 cm(-1) where the onsets of the 2 (3)Pi(1) and 3 (3)Sigma(+)(1) <- X (1)Sigma(+) transitions are located, the upper electronic states and the vibrational quantum numbers were assigned. Perturbations of 2 (3)Pi(1) and 3 3 Sigma(+)(1) and 2 (1)Pi-3 (3)Sigma(+)(1) have been identified by the observed level shifts.