It looks like the Graham amendment for increased border security has been stopped for now. Forbes has some of the details:

It started with an end-run by Republicans to pass some of the most popular elements of President Bush's failed immigration bill, including a plan to increase security along the southern border.

Democrats liked the money but objected to such GOP proposals as allowing law enforcement officers to question people about their immigration status and cracking down on those who overstay their visas.

The move put political pressure on Senate Democrats. They killed Sen. Lindsey Graham's plan on a 52-44 procedural vote, but Majority Leader Harry Reid, D-Nev., immediately countered with a pared-down proposal containing only the border security funds. Then Sen. John Cornyn, R-Texas, a Bush ally, killed that effort.

Dying with the proposals could be hopes for seizing "operational control" over the U.S.-Mexico border with additional Border Patrol agents, vehicle barriers, border fencing and observation towers.

Looking at the roll call vote to block the Graham amendment, three Republicans voted to block: Cochran (MS), Stevens (AK), and Voinovich (OH). No Democrat voted to continue with the Graham amendment.

Update: More details at the OC Register. Reid's objection to block the amendment:

But in the end, Reid used a procedural maneuver to sideline the Republican amendment as well as his addition to it. He objected to the amendment because its provisions, he said, would be making legislative policy on a spending bill, something the Senate does not do.

The LAT describes the debate as "tart"! It seems as though AgJobs might have come up, but this move was scuttled:

In the midst of the debate, it appeared that AgJobs, a five-year work program for farm workers championed by Sen. Dianne Feinstein (D-Calif.) and Sen. Larry E. Craig (R-Idaho), might come up. When it was clear that would not happen, the two senators appealed to their colleagues to consider the measure.

Although it also sounds as though Reid is willing to try to bring AgJobs back.

How did Cornyn block Reid's attempt to bring just a $3 bill. bill to the floor? He objected to Reid's offer of a unanimous consent agreement to bring only the funding portion back. The LAT has more on how Sen. Chambliss (R-GA) objects to AgJobs as long as it allows for some pathway to permanent residency. Or so he presently says...

So do some Republicans still hope they can pressure Democrats into accepting a border-security (only) bill? It would seem as though Cornyn's blocking of border security-only funding would be one attempt to put some pressure on some Democrats or at least Reid...

UPDATE 7/26: It looks like the $3 billion funding is back.