Byrne picked up the injury in Wales' 37-25 loss at the hands of New Zealand two weeks ago.
But the severity of the injury was not discovered until now after Byrne failed to attend two consultations.
The full-back will now miss the crucial Heineken Cup double header with Munster, as well as Christmas derby matches with the Scarlets and Cardiff Blues.
"We're now in a position to confirm the severity of the injury Lee suffered in the recent test match against New Zealand, which kept him out of the Edinburgh fixture last weekend," said Ospreys physio Chris Towers.
"Further investigations yesterday confirmed a fracture at the base of the thumb, and in light of this he is unlikely to be available for selection for an estimated period of five weeks from now."