According to the story, a California court ordered the parties to mediate the dispute, and they signed an agreement in which the Winklevosses would give up ConnectU for cash and a share in Facebook stock.
The Winklevosses later tried to pull out of the deal, alleging that Facebook had undervalued its stock, thus rendering the agreement as having been procured fraudulently.
A judge ruled against them and forced the settlement to go through, but the Winklevosses appealed.
U.S. 9th Circuit Court of Appeals Chief Judge Alex Kozinski, ruled against them.
"At some point, litigation must come to an end," Kozinski concluded. "That point has now been reached."
The brothers will not to appeal their case to the U.S. Supreme Court.