Ocean current measurements play a crucial role in aiding our understanding of ocean dynamics and circulation systems. Traditional methods, such as drifters and ocean buoys, are sparsely distributed and of limited effectiveness due to the nature of the marine environment and high operating expenses. Distributed acoustic sensing (DAS) is an emerging technology using submarine optical-fiber (OF) cables as dense seismo-acoustic arrays, offering a new perspective for ocean observations. Here, in situ observations of ocean surface gravity waves (OSGWs) and ocean currents by DAS were made along a pre-existing 33.6 km seafloor OF cable. The average current velocity and water depth along the cable were determined from observed OSGW-induced seafloor noise (0.05-0.2 Hz) using ambient-noise interferometry and frequency-domain beamforming. Variations in current velocity were derived at high spatiotemporal resolution using the frequency-domain waveform-stretching method. The inverted current velocity was verified by nearby ocean buoy observations and forecasting results. The observations demonstrate the effectiveness of DAS-instrumented OF cables in monitoring ocean currents.