WASHINGTON — Cheered by conservative supporters, President Bush gave a push Monday to a constitutional amendment to ban same-sex marriage as the Senate opened debate on an emotional, election-year measure that has little chance of passing.

"Our policies should aim to strengthen families, not undermine them," Bush said in a speech. "And changing the definition of marriage would undermine the family structure."

All Senate Democrats, except Ben Nelson of Nebraska, oppose the amendment, and critics say Bush's efforts are primarily aimed at energizing conservative voters for the November elections. Together with moderate Republicans, the Democrats are expected to block a yes-or-no vote, killing the measure for the year.

Sen. Wayne Allard, R-Colo., who sponsored the measure, acknowledged that politics played a part in the timing of the debate, but for a different reason: to force senators to take a stand and answer for their votes on the campaign trail.

"We ought to have a vote on the amendment every year," Allard said.

The amendment would prohibit states from recognizing same-sex marriages. To become law, it would need two-thirds support in the Senate and House, and then would have to be ratified by at least 38 state legislatures.

Sen. Hillary Clinton said the discussion of gay marriage in Congress this week is a priority for the White House and the Republican majority but not for her constituents.

Most people "worry about everything from terrorism to gas prices to the cost of health care to energy independence," Clinton, D-N.Y., said Monday at a fund-raising luncheon.

The White House played down the significance of the 10-minute presidential event, saying Bush was simply speaking out on an issue being debated on Capitol Hill. Press secretary Tony Snow said the president was not personally lobbying senators to pass the amendment.

Bush also pressed for it in his radio address on Saturday.

"I'm not sure this is a big driver among voters," Snow said.

Bush said a constitutional amendment is needed because laws that state legislatures have passed defining marriage as being between a man and a woman are being overturned by a few judges.

"When judges insist on imposing their arbitrary will on the people, the only alternative left to the people is an amendment to the Constitution — the only law a court cannot overturn," the president said.

Bush also rebuffed critics who argue that the amendment conflicts with the GOP's opposition to government interference and the importance of states' rights.