The circulation of water masses in the northeastern North Atlantic Ocean has a strong influence on global climate owing to the northward transport of warm subtropical water to high latitudes1. But the ocean circulation at depths below the reach of satellite observations is difficult to measure, and only recently have comprehensive, direct observations of whole ocean basins been possible2, 3, 4. Here we present quantitative maps of the absolute velocities at two levels in the northeastern North Atlantic as obtained from acoustically tracked floats. We find that most of the mean flow transported northward by the Gulf Stream system at the thermocline level (about 600 m depth) remains within the subpolar region, and only relatively little enters the Rockall trough or the Nordic seas. Contrary to previous work5, 6, our data indicate that warm, saline water from the Mediterranean Sea reaches the high latitudes through a combination of narrow slope currents and mixing processes. At both depths under investigation, currents cross the Mid-Atlantic Ridge preferentially over deep gaps in the ridge, demonstrating that sea-floor topography can constrain even upper-ocean circulation patterns.