Sudan's President Omar al-Bashir threatened to invade the capital of South Sudan and topple its government, in a widening fight over an oil-rich border region that is turning into a test of economic survival for the east African neighbors.

Mr. Bashir made the comments in a rally of his troops, who are now engaged on at least three fronts with South Sudan, including the oil hub of Heglig, after launching two fresh assaults the day before. "Heglig isn't the end, it is the beginning,...