We present the results of 44 new heat flow stations which were taken in 1984 and 1989 in profiles across the Goban Spur and Galicia Bank continental margins (NW Atlantic Ocean). Simple extensional models indicate that the heat flow across both these Early Cretaceous rifted margins should increase from values of 45-50 mW/m2 over oceanic crust to 65-80 mW/m2 on the continents. The rate of this increase should help to constrain the mechanism (simple versus pure shear) by which the upper, more radiogenic continental crust is thinned. Measurements across Goban Spur increase from values of 40-45 mW/m2 over oceanic crust to 50-55 mW/m2 near the continental shelf. They follow the predicted trend for pure-shear rifting, but only if the value of upper crustal radiogenic heating is low (1-2-mu-W/m3). Otherwise, they would require the upper crust to thin more rapidly than the total crustal thickness, as with a lower plate, simple-shear margin. Measurements across Galicia Bank show a very different pattern, with similar values over oceanic crust but much lower values (30-35 mW/m2) nearer land. This is difficult to reconcile with any simple, single rifting event but is more compatible with an origin as a pure-shear or lower plate rather than upper plate margin. We also note that oceanic values of heat flow require asthenospheric temperatures 100-degrees-C lower than normal for both margins. This indicates that the triple junction in existence between these margin segments during the breakup of Iberia, Europe, and North America was not the site of a major mantle plume.