The Western Valley is one of the passages across the Iceland–Scotland Ridge through which a strong overflow of cold, dense water has been thought to feed the deep limb of the Atlantic Meridional Overturning Circulation (AMOC), but its strength has not been known. Based on a field experiment with instruments moored across the valley, we show that this overflow branch is much weaker than previously thought and that this is because it is suppressed by the warm countercurrent in the upper layers.