For the Republican elites who overwhelmingly favor immigration reform, this is a grim prospect. "This is the fight for the soul of the party," said John Feehery, a former top aide to House Speaker Dennis Hastert and House Majority Leader Tom DeLay. "Marco Rubio and Paul Ryan get it. Ted Cruz doesn't get it. It's the defining struggle for the Republican Party, and right now the good guys are losing."

The debate over Republicans' approach to immigration has largely focused on politics—on whether and how the party will be able to woo Hispanic voters in the next presidential election. But the intra-party psychodrama is bigger than that. It's about whether the pragmatists can seize the reins of the Republican Party, or whether the angry, oppositionist, populist strain retains control. (Feehery calls them "the haters," and sees them as the heirs to the Know-Nothings who tried to keep out his Irish ancestors.)

The consequence, these more moderate Republicans fear, will be a GOP that remains the party of Sarah Palin and Donald Trump, one that is content to excite the passions of an irate base without even pretending to propose solutions to the country's problems.

"We will not be a national governing party for a long, long time if we turn our backs on this chance to pass immigration reform. It's just that simple," said John Weaver, the former John McCain and Jon Huntsman strategist, calling the prospect "depressing."

By rejecting immigration, he said, the Republicans in the House are sending a message that they're not interested in being part of the solution. "If you only have to worry about your right flank—you don't have to worry about a general election, don't have to worry about governing—that's a pretty easy gig, isn't it? What the hell is the point?"

Among Republican power brokers, support for immigration reform—meaning a comprehensive bill that legalizes the undocumented, expands legal immigration, and increases border security—is virtually universal. The U.S. Chamber of Commerce, the business lobby that spent more than $30 million against Democratic candidates in 2012, has endorsed the Senate proposal. Karl Rove, the last strategist to win a presidential election for the party, has repeatedly devoted his Wall Street Journal column to urging reform, and George W. Bush recently emerged from his post-presidential seclusion to do the same. In a post-election "autopsy" commissioned by the Republican National Committee, the only policy prescription was this: "We must embrace and champion comprehensive immigration reform." The committee that drafted the report included former Bush Press Secretary Ari Fleischer and prominent GOP strategists Henry Barbour and Sally Bradshaw.

A group called Republicans for Immigration Reform is headed by Bush's commerce secretary, Carlos Gutierrez; its board includes fellow Bush Cabinet members Spencer Abraham and Margaret Spellings, as well as Charlie Spies, who last headed the pro-Romney super PAC Restore Our Future, and Malek, a prominent GOP fundraiser since the Nixon Administration. The American Action Forum, whose president is a former political director for the National Republican Congressional Committee, is airing ads boosting immigration reform; the group is backed by former Senator Norm Coleman and former Bush and McCain economist Douglas Holtz-Eakin. FWD.us, the pro-reform group backed by Facebook's Mark Zuckerberg, has a Republican subsidiary called Americans for a Conservative Direction whose board that includes Haley Barbour, the onetime Mississippi governor and RNC chairman, and Dan Senor, the former Bush and Romney adviser. The Southern Baptists, normally one of the most conservative religious groups, are among several that have aggressively advocated for an immigration solution that includes citizenship for the undocumented.