WASHINGTON -- President George W. Bush vetoed a bipartisan bill on Wednesday that would have dramatically expanded a government health insurance program for children.

It was only the fourth veto of Bush's presidency, and one that some Republicans feared could carry steep risks for their party in next year's elections.

The Senate approved the bill with enough votes to override the veto, but the margin in the House fell short of the required number.

The White House sought as little attention as possible, with the president wielding his veto behind closed doors without any fanfare or news coverage.

The State Children's Health Insurance Program is a joint state-federal effort that subsidizes health coverage for 6.6 million people.

Most of those covered are children from families that earn too much to qualify for Medicaid but not enough to afford their own private coverage.

Democrats, with significant support from Republicans, passed the legislation to add $35 billion over five years to allow an additional 4 million children into the program.

The increase would be funded by raising the federal cigarette tax by 61 cents to $1 per pack.

Despite the veto, White House spokeswoman Dana Perino said Wednesday that Bush "does want to work with members of Congress to see if we can find common ground."

Asked to elaborate, she said, "I think that we would like to see Congress look at a bill that would make sure that the neediest children are taken care of first."