Measurements of velocity of oceanic currents are crucial for evaluating heat and mass transport, monitoring ocean dynamics, and understanding the ocean's role in climate. Reciprocal transmissions of acoustic signals provide a way to measure path-averaged current velocity in the ocean with high temporal resolution, on a large scale, and over extended periods of time. This approach is based on measurements of acoustic nonreciprocity, which is defined as differences between travel times or other acoustic quantities corresponding to sound propagation in opposite directions between two points. Acoustic nonreciprocity, which vanishes in motionless media, is a direct, sensitive measure of fluid motion, and is insensitive to uncertainties in the sound speed field and geometry of the experiment. The use of active reciprocal transmissions for ocean remote sensing applications is limited by capital and operational costs associated with low-frequency acoustic sources as well as concerns about potential impacts of transmissions on marine life. Underwater acoustic noise interferometry offers an alternative to active remote sensing by replacing probing signals generated by a dedicated source with two-point cross-correlations of diffuse ambient noise. Theory predicts that the cross-correlation function of diffuse noise measured at two locations in a generic inhomogeneous moving medium results in approximations to Green's functions describing sound propagation in opposite directions between the two measurement points; positive and negative lags correspond to propagation in opposite directions. Thus, two-point correlation functions allow one to quantify flow-induced acoustic nonreciprocity. In this paper, we use the data obtained in a noise interferometry experiment in the Straits of Florida to investigate the feasibility of passive remote sensing of currents in the ocean. We present the first experimental demonstration that acoustic nonreciprocity induced by oceanic currents can be measured by noise interferometry and that the current velocity can be retrieved from the acoustic noise cross-correlations.