However, the article goes on the caution that Lieberman is unpredictable (which is a euphemistic way of saying he has a knack for lying):

“Lieberman keeps assuring Reid that he’s OK,” said one source. “But he’s one of those characters — you never know with Joe.

And we still have this:

“Sen. Lieberman has made it clear that he will vote for the motion to proceed to the healthcare bill but will oppose cloture on a final bill if it contains a public option because he believes that it would worsen our national debt problem,” said Lieberman aide Marshall Wittmann.

So it's far from a done deal, but definitely a step in the right direction.

Update:

There is some confusion in the comments over what has actually changed. Previously, Lieberman has claimed he will allow debate of the bill, but prevent a final vote. He is still saying this publicly (see quote above) but the article's claim is that he is privately assuring Reid he will allow a final vote. The article could be false, and as BennyToothpick pointed out in the comments, what you think of this news depends on how much you trust TheHill's unnamed sources. However, if true it probably means that the bill can get out of the Senate with the opt-out Public Option.