One way for this to happen would be for the presiding officer -- most likely Vice President Biden or Senate president pro tempore Pat Leahy -- to declare it unconstitutional, either in whole or in part (for example, for just judicial and executive nomination). That's a break with Senate precedents, by which the body as a whole has to decide constitutional issues. A Republican Senator would likely appeal, and then a Democratic Senator would move to table the appeal, which would then go to a majority vote of the chamber. If a majority tables the appeal, the filibuster is effectively amended or ended. Alternately, a Democratic Senator could submit a point of order that the filibuster is unconstitutional (in whole or in part), after which the presiding officer would submit the question to the Senate as a whole. Usually, such a question would be subject to debate, but the presiding officer could break precedent and declare it undebatable. A Republican Senator would likely appeal, and then another Democratic Senator would motion to table that appeal, which would go to an immediate vote. If a majority votes to table, the Senate would proceed to vote on the original motion to declare the filibuster either wholly or partially unconstitutional. If a majority votes yes, the filibuster is amended or ended.
At 10:30 am eastern, Senate Majority Leader Harry Reid (D-Nev.) went to the Senate floor and began the process of changing the chamber's rules on filibusters of judicial and administrative nominees. "The American people believe that the Senate is broken," Reid said. "And I believe that the American people are right."
Senate Minority Leader Mitch McConnell (R-Ky.) followed moments later with a lengthy speech about his opposition to the Affordable Care Act. It was an odd non sequitur, though for congressional Republicans, everything is always about "Obamacare."
Regardless, it would appear the so-called "nuclear option," years in the making, has been set in motion. The procedural tactics can get a little complicated, but if you're watching C-SPAN's live coverage, keep Dylan Matthews' review handy (via Brad Plumer).
As I type, the Senate is re-voting on Patricia Millett's D.C. Circuit nomination, filibustered exactly three weeks ago. The filibuster is likely to work, at which point Democrats will take the next step in changing the rules. (Technically, this vote is on "a motion to proceed to the motion to reconsider the vote by which cloture was not invoked on the Millett nomination," which is one of those easy-to-understand sentences that makes legislative procedure so wildly entertaining.)
Senate Pro Tem Pat Leahy (D-Vt.), who had been a skeptic of the "nuclear option," is presiding over the Senate.
For Senate reform advocates, tired of watching members come close to the "nuclear" line without crossing it, today is a pretty amazing day.