Underwater acoustic (UWA) channels are wideband in nature due to the small ratio of the carrier frequency to the signal bandwidth, which introduces frequency-dependent Doppler shifts. In this paper, we treat the channel as having a common Doppler scaling factor on all propagation paths, and propose a two-step approach to mitigating the Doppler effect: 1) nonuniform Doppler compensation via resampling that converts a "wideband" problem into a "narrowband" problem and 2) high-resolution uniform compensation of the residual Doppler. We focus on zero-padded orthogonal frequency-division multiplexing (OFDM) to minimize the transmission power. Null subcarriers are used to facilitate Doppler compensation, and pilot subcarriers are used for channel estimation. The receiver is based on block-by-block processing, and does not rely on channel dependence across OFDM blocks; thus, it is suitable for fast-varying UWA channels. The data from two shallow-water experiments near Woods Hole, MA, are used to demonstrate the receiver performance. Excellent performance results are obtained even when the transmitter and the receiver are moving at a relative speed of up to 10 kn, at which the Doppler shifts are greater than the OFDM subcarrier spacing. These results suggest that OFDM is a viable option for high-rate communications over wideband UWA channels with nonuniform Doppler shifts.