President Bush will unilaterally obligate the United States. Neither the Senate (pursuant to its constitutional role in ratifying treaties) nor the House will have any say. While the reduction of the Senate and House to ciphers in national security matters through executive agreements flouts the Constitution, it has become as commonplace as the rising and setting of the sun. It is no longer news.
Alexander Hamilton, who advocated a muscular executive, voiced the unambiguous consensus among the Founding Fathers that an unchecked president in foreign affairs would be dangerous - even with George Washington at the helm.
Accordingly, the treaty power was divided between the president and Senate in Article II, section 2, clause 2. Hamilton elaborated in Federalist 75: "The history of human conduct does not warrant that exalted opinion of human virtue which would make it wise in a nation to commit interests of so delicate and momentous a kind, as those which concern its intercourse with the rest of the world, to the sole disposal of a magistrate created and circumstanced as would be a president of the United States."