The two-photon absorption (TPA) coefficient β and the nonlinear index of refraction n2 for bulk cuprous oxide (Cu2O) direct gap semiconductor single crystal have been measured by using a balance-detection Z-scan single beam technique, with an excellent signal to noise ratio. Both coefficients were measured at 790 nm using a 65 fs laser pulse at a repetition rate of 90.9 MHz, generated by a Ti:Sapphire laser oscillator. The experimental values for β were explained by using a model that includes allowed-allowed, forbidden-allowed, and forbidden-forbidden transitions. It was found that the forbidden-forbidden transition is the dominant mechanism, which is consistent with the band structure of Cu2O. The low value for β found in bulk, as compared with respect to thin film, is explained in terms of the structural change in thin films that result in opposite parities of the conduction and valence band. The n2 is also theoretically calculated by using the TPA dispersion curve and the Kramers-Kronig relations for nonlinear optics.