Europe’s highest court, the European Court of Justice, has fined Greece 2m euros ($2.8m; £1.7m) for failing to recover state aid from Olympic Airways.
The case dates back to 2002 and relates to 41m euros of state aid which was given to the airline as part of an unsuccessful restructuring.
Since the case began the carrier has been renamed Olympic Airlines.
Athens has one month to comply or it will face additional fines of 16,000 euros a day until it pays up.
The money will be paid into the EU Budget.
In 2005 the European Commission found the Greek government guilty of giving illegal state aid to Olympic and demanded it reclaim the money.
Greece said in court that much of the aid had been paid back, but the court found that the proof provided by Greece to show that it had already recovered the aid was inappropriate.
The European Commission approved the privatisation of the airline last year on condition that Athens reimburse aid given to the firm, which has been state owned since 1975.