The amount of magmatism that accompanies the extension and rupture of the continental lithosphere varies dramatically at rifts and margins around the world. Based on widely spaced geophysical transects, some margins are known to preserve a transition from magmatically robust to magmatically starved rifting along strike, but the nature of the transition is unknown. Wide-angle seismic data from the Black Sea provide the first direct observations of such a transition and show that it is abrupt, occurring over only ~20–30 km, and coincides with a transform fault. This abrupt transition cannot be explained solely by gradual along-margin variations in mantle properties, since these would be expected to result in a smooth transition from magma-poor to magma-rich rifting over hundreds of kilometers. We suggest that the abruptness of the transition results from the development of three-dimensional (3-D) melt migration due to along-strike variations in extension and thus the thickness of the lithosphere at the time of rifting. Localized magmatic addition attributed to melt focusing has been observed in modern mid-ocean ridges and active rift environments, but here we show that such processes can also produce abrupt along-strike changes from magma-poor to magma-rich rifting.