In theory.
In practice, Presidents increasingly unilaterally commit the country to war, with Libya being the most recent example.
Glenn Greenwald discusses the flimsy legal basis and historical context of Presidential assumption of authority to make war. Matt Yglesias makes the very valid political point that the President is able to assume Congressional power in this area because Congress wants him to.
Both of those are fine discussions but there are some other interesting implications here:
(1) Presidential war-making authority makes war more likely. As a single actor, Presidents have a lower threshold for decision making than two houses of Congress. In addition Presidents feel more responsible for the consequences of inaction.
(2) Presidential war-making is facilitated by (and, therefore, probably promotes) an excessively large military establishment. That is, because the President can bomb Libya without raising troops or procuring new weapons the President is freer to do so. Similarly, to maintain this freedom Presidents will want a larger military than would otherwise be optimal.
(3) We can't, institutionally, admit that a portion of our constitution is no longer operative. Thus, we are over time likely to conclude that Presidents may, notwithstanding the language of the Constitution commit the United States to war.
(4) The logic behind the Congressional abdication of responsibility here applies elsewhere and thus Congress will increasingly cease to be a meaningful check on the Presidency for all but the most unpopular actions.