But it comes with the condition that U.S. troops start withdrawing from Iraq by October 1.
Focus On Terrorism
The House, controlled by the Democratic Party, passed the bill by 218 to 208.
"By placing an unacceptable strain on our military, this war is undermining our ability to protect the American people," House Speaker Nancy Pelosi said. "Instead of making the American people safer, the war in Iraq has weakened our ability to protect our nation from the threat posed by international terrorism."
Pelosi urged Bush "to sign the bill so that we can focus on winning the war against terrorism, which is the real threat to the American people."
The Senate is expected to pass the same bill by the end of this week.
In a highly symbolic move, Democrats say they want the bill to arrive on Bush's desk on May 1, the fourth anniversary of his announcement on aboard the deck of the "USS Abraham Lincoln" that major combat operations in Iraq had ended.
Democrats say they are encouraged by the release of a new poll that shows that a majority of U.S. citizens side with them on the issue and believe victory in Iraq is no longer possible.
Majority Of Americans Favor Deadline
According to the latest NBC News/"Wall Street Journal" poll, 56 percent of those questioned said they agreed with setting a deadline for troop withdrawal. Meanwhile 37 percent sided with Bush in opposing the move.
John Boehner, the Republican minority leader in the House of Representatives, was one of those on April 25 who made the case against setting a withdrawal deadline.
"We can walk out of Iraq, just like we did in Lebanon, just like we did in Vietnam, just like we did in Somalia, and we will leave chaos in our wake," Boehner said.
The commander of U.S. forces in Iraq, General David Petraeus, also met lawmakers to argue against the bill.
Veto Showdown Coming
Bush has repeatedly threatened to veto the bill.
"Congress's failure to fund our troops on the front lines will mean that some of our military families could wait longer for their loved ones return from the front lines, and others could see their loved ones headed back to the war sooner than they need to. That is unacceptable to me, and I believe it is unacceptable to the American people," Bush said earlier this month.
Though the Democrats control both houses of Congress, they do not have enough votes to overrule a presidential veto.
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