Fresh off a breakthrough win paving the way for scrapping the military’s “don’t ask, don’t tell” policy, gay rights advocates in the House said they have new momentum for their next act: a bill banning employers from discriminating against gay and transgender people.

But the measure, the Employment Non-Discrimination Act, causes heartburn for politically imperiled Democrats looking to duck divisive votes in a toxic environment for incumbents.

Those moderate lawmakers and staff are quietly angling to shelve the bill, arguing that success on “don’t ask, don’t tell” would be the second major win for the gay community this Congress, following enactment last year of hate-crimes legislation.

And many assume their leadership is on the same page. “After ‘don’t ask, don’t tell,’ it’s dead as disco,” one senior Democratic aide said.

Whatever hope they have of burying the debate, however, is likely to collide with a hard reality: Speaker Nancy Pelosi (D), hailing from San Francisco, the birthplace of the gay rights movement, has pledged to move ENDA this year.

Rep. Barney Frank, ENDA’s leading Congressional proponent, said he received assurances from Pelosi and other leaders that the vote to roll back “don’t ask, don’t tell” wouldn’t preclude consideration of the discrimination ban. The Massachusetts Democrat, one of three openly gay Members, said that while ENDA now would undoubtedly get “pushed back” somewhat, he spoke with House leaders after Thursday’s vote to overturn the ban on openly gay individuals serving in the military and was confident that Pelosi would fulfill her promise to take up ENDA before the end of this Congress. “The Speaker’s still very committed to it,” he said.

Pelosi made her commitment clear last month, when she told gay rights supporters on a conference call that she intended to put both the “don’t ask, don’t tell” repeal and ENDA to a vote in this Congress. “It’s not one or the other,” Pelosi said, according to one account of the call reported by the Advocate. Appearing later that month at a rally celebrating slain gay rights leader Harvey Milk, Pelosi made a plea for the nondiscrimination law. “Why is it even an issue that we’re talking about discrimination in the workplace in this day, this age, this century? Get this over with,” she said.

A Democratic leadership aide confirmed that Pelosi remains intent on pursuing the measure this year.

One option that leaders are eyeing to limit the political fallout while still delivering on the vote would be to delay consideration until a lame-duck session after the midterm elections. Gay rights advocates said such a move would be consistent with Pelosi’s pledge. “The sooner, the better,” said Laurie Young, who heads the department of public policy and government affairs at the National Gay and Lesbian Task Force. “But this Congress doesn’t end until the next one is sworn in.”

And Frank said that based on his recent discussions with House leaders, “the timing depends on a lot of other factors that can’t be determined.”

In the meantime, the whipping operation to bring Members on board continues, and Frank said it was “going well.” The focus of that push in recent weeks has been to lock down support for beating back an anticipated Republican motion to strike the language extending nondiscrimination protections to transgender workers. Frank said he believes he now has the votes to do so. “If it’s a straight motion like that, yes,” Frank said, declining to comment on what other, more challenging variations on the transgender language he was girding against.

House leaders originally had worked to put ENDA on the floor before the “don’t ask, don’t tell” repeal. But Frank said it might be easier to woo some Members — presumably Republicans — after the primary season has passed.

Rep. Tammy Baldwin, another openly gay member helping to round up votes for the measure, said she remained hopeful that the bill could come to the floor after the Memorial Day recess. And the Wisconsin Democrat downplayed the notion that Members would be wary of voting on another gay rights issue before the midterms. “If you think about it … 75 percent of the population supports repeal of ‘don’t ask, don’t tell,’ so while there may be a handful of Members for whom that was a difficult vote, for the vast majority, they’re stepping right in line with the mainstream American public and their views,” she said. “So I just don’t buy that.”

But a swath of moderates have consistently voiced angst. Rep. Bobby Bright, a freshman Democrat who faces a tough fight this fall in an Alabama district that favored Sen. John McCain (R-Ariz.) in the 2008 presidential race by 26 points, said he would prefer not to have to consider the issue this year.

“It would be a difficult vote for someone like me,” said Bright, who added that he was undecided on how he would vote and had not yet been approached by the whip operation.

Bright, who was among the 26 Democrats who opposed repeal of “don’t ask, don’t tell,” said he was wary of any more politically challenging votes.

“All I know is that since I’ve been here, we’ve been doing some heavy lifting, and I don’t see it letting up,” he said.

Added another moderate: “If you’re talking about retaining the majority, unfortunately, that type of bill has been used against us. People don’t understand it. That’s the perception, and we have to deal with the perception, not the reality, sometimes.”

But the chamber’s newest Democrat, Rep. Mark Critz, who won a May 18 special election in the largely rural, southwestern Pennsylvania district that the late Rep. John Murtha represented, opposed the “don’t ask, don’t tell” amendment, but said he was open to perhaps supporting ENDA.

“I’m going to have to look at it and see what I can support, what I can’t support, and then work with leadership on that,” said Critz, who noted that he voted “no” on “don’t ask, don’t tell” out of deference to military leaders, who indicated they would prefer to wait until a Pentagon review is complete.