Genuine full-duplex operation requires effective mitigation of self-interference (SI) due to simultaneous transmission and reception at the same frequency band. In addition to its well-known harmful effect on signals of high peak-to-average power ratio, nonlinear behavior of a power amplifier (PA) complicates SI cancellation and induces spectral regrowth. We introduce a digital predistortion architecture that linearizes the response of the PA in a full-duplex transceiver. Specifically, we propose a two-step procedure to estimate the predistorter parameters and the SI canceller coefficients. Different from a direct application of conventional predistorters, the proposed architecture does not need an extra RF chain to estimate the PA response and exploits the inherent SI signal instead. Finally, simulation results show that the proposed scheme is able to increase significantly the signal-to-interference-plus-noise ratio at the transceiver output and to reduce out-of-band emissions when compared to linear and nonlinear cancellation without predistortion.