The president was making further remarks Monday in favor of the amendment as the Senate opened three days of debate on it.
All but one of the Senate Democrats - the exception is Ben Nelson of Nebraska - oppose the measure and, with moderate Republicans, are expected to block an up-or-down vote, killing the measure for the year.
Democrats say the amendment is a divisive bow to religious conservatives, and point out that it conflicts with the Republican Party's opposition to big government interference.
Fueled by election-year politics, the gay marriage issue is the most volatile Congress will consider as it returns from a weeklong Memorial Day recess.
Other legislation has better chances for success, particularly a record-size emergency spending bill to continue U.S. military operations in Iraq and Afghanistan and provide hurricane relief along the Gulf Coast, according to the AP.
Meanwhile, the Senate Judiciary Committee will hold a hearing Tuesday on government surveillance of journalists who publish classified information, the result of probes into published reports on secret prisons overseas and the Bush administration's domestic wiretapping program.
An election-year debate on the constitutional amendment to define marriage as a union between a man and a woman was never in doubt, however doomed the legislation. As Republicans geared up to defend their majorities in the House and Senate, conservative groups earlier this year let them know that they were dissatisfied with the Republicans' efforts on several social issues, including gay marriage.
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