The Senate looks to be on track to pass its immigration overhaul today, getting out of town for the July Fourth recess a day early.

After two more largely anticlimactic votes, it appears to be all systems go for a final passage vote Thursday afternoon, sending a bill that would significantly rewrite immigration and border security laws for the first time since 1986 to the House of Representatives, which plans to take a decidedly different approach.

In the second of the three early Thursday votes, senators voted, 68-32, to limit debate on the underlying bill. That was well clear of the 60 votes required to invoke cloture under Senate rules, but short of the 70 votes that supporters had said would put pressure on House Republican leadership to take up the bill.

(Also on Roll Call: 70 Votes for Immigration Seems a Stretch, and May Be Moot | Hawkings Here)

Ahead of the morning votes, Minority Leader Mitch McConnell made the not unexpected announcement that he would oppose final passage. The Kentucky Republican has largely stayed in the shadows, deferring to other GOP senators on the immigration question and declining to engage in public procedural spats with Majority Leader Harry Reid, D-Nev., on the issue with the bill on the floor.

“It’s with a great deal of regret, for me at least, that the final bill didn’t turn out to be something I can support,” McConnell said. “The reason is fairly simple. As I see it, this bill just doesn’t meet the threshold test for success that I outlined at the start of this debate. It just doesn’t say to me, at least, that we’ve learned the lessons of 1986, and that we won’t find ourselves right back in the same situation we found ourselves in after that reform.”

There’s likely to be few, if any, changes from that tally on the final passage vote, which appears on track to take place around 4 p.m. Thursday afternoon.

Reid said Thursday morning that he hoped to set up the getaway vote as soon as possible, before adding “but if people want to delay it they can.” Indeed, if senators wanted to take all the time available to them, the measure would not be up for final passage until Friday.

Reid also called for senators to vote from their desks on final passage of the immigration bill, in a nod to the importance of the measure. That’s one move for which the majority leader doesn’t actually need unanimous consent.