Winklevoss Twins LOSE AGAIN -- Must Pay Lawyers $13 MILLION

The legal losing streak continues for Cameron and Tyler Winklevoss ... who've been ordered to pay their former lawyers $13 MILLION in fees after losing yet another appeal in their neverending Facebook saga.

The Winklevii have been at war with Quinn Emanuel Urquhart & Sullivan -- the firm that represented them in the case portrayed in "The Social Network." The boys claim the firm committed malpractice by leaking information about their $65 million settlement with Mark Zuckerberg ... and they shouldn't have to pay them a single cent.

But an arbitration panel ruled against the twins ... so they took the case all the way up to a New York appeals court ... which just dished out some more bad news for the Winks ... they lost

Now, the Winklevii have no choice but to pay the fee ... or else the firm can take steps towards seizing their assets.