In this paper, we address the problem of combating the effect of the nonlinearities of the transmit-receive chain in solid-state radars, which are typically generated by the power amplifier in transmission. This issue is of particular importance in meteorological radar applications, as they require transmit pulses featuring very low autocorrelation sidelobes, which are significantly raised due to nonlinearities. In this work, we describe an adaptive predistortion design method based on an iterative approach in the spectral domain. The proposed method has been validated by means of hardware using a customized software radio and driving an amplifier into saturation. Two different ideal, highly performing pulse compression waveforms, expressly designed for weather radar applications, were sent to its input. The experimental results demonstrate the benefits of the proposed approach to design the digital predistortion module.