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Abstract The cross‐correlation of a diffuse or random wavefield at two points has been demonstrated to recover an empirical estimate of the Green's function under a wide variety of source conditions. Over the past two decades, the practical development of this principle, termed ambient noise interferometry, has revolutionized the fields of seismology and acoustics. Yet, because of the spatial sparsity of conventional water column and seafloor instrumentation, such array‐based processing approaches have not been widely utilized in oceanography. Ocean‐bottom distributed acoustic sensing (OBDAS) repurposes pre‐existing optical fibers laid in seafloor cables as dense arrays of broadband strain sensors, which observe both seismic waves and ocean waves. The thousands of sensors in an OBDAS array make ambient noise interferometry of ocean waves straightforward for the first time. Here, we demonstrate the application of ambient noise interferometry to surface gravity waves observed on an OBDAS array near the Strait of Gibraltar. We focus particularly on a 3‐km segment of the array on the continental shelf, containing 300 channels at 10‐m spacing. By cross‐correlating the raw strain records, we compute empirical ocean surface gravity wave Green's functions for each pair of stations. We first apply beamforming to measure the time‐averaged dispersion relation along the cable. Then, we exploit the non‐reciprocity of waves propagating in a flow to recover the depth‐averaged current velocity as a function of time using a waveform stretching method. The result is a spatially continuous matrix of current velocity measurements with resolution <100 m and <1 hr.