That is built on the recognition that Democrats have plenty of conservative-state senators in their caucus who have long been allies of the National Rifle Association and who are likely to be punished politically if they waver from that support.
As recently as 2009 — when Democrats held 59 Senate seats — Republicans advanced gun-rights measures that allowed firearms in national parks and on Amtrak trains. They passed overwhelmingly as half the Democratic caucus joined the Republicans in support of the looser restrictions.
If the gun measure passes the Senate, its future in the House remains uncertain.
Rep. Peter King, R-N.Y., a moderate, has suggested that he would offer the same legislation in that chamber, but the House has much stricter rules, and such a debate would need to have leadership support.
House Speaker John Boehner, R-Ohio, politically weakened for repeatedly allowing legislation to pass with a majority of Republicans opposed, shows no eagerness for such a divisive debate.
Immigration overhaul would appear to be on surer political footing, even though it is not as far along as gun legislation.
Unlike his work opposing the gun measure, Graham is leading the effort to defend the tentative agreement that the eight senators have crafted and is under final review.
"We'll try to beat back amendments that are designed to kill the bill, except good ideas that make it better, and fight for the bill. That's all you can do," Graham said.
The plan is to unveil the immigration measure by early next week and hold a public hearing before the Judiciary Committee shortly thereafter.
After that, the panel is expected to consider many amendments — all of which will give an early indication of how likely it is that the new coalition will hold together. The last effort at a comprehensive immigration package, in June 2007, fell apart amid a flurry of amendments from the liberal and conservative wings of each party.