The French frigate was escorting another tanker and had attracted at least 17 ships in its wake seeking protection but the Liberian tanker had stayed on its course.
The pirates freed a Greek freighter on Friday, leaving 17 ships still in their hands despite foreign warships stepping up their efforts to fight the piracy.
The pirates are still holding the giant Saudi oil tanker Sirius Star which they hijacked on November 15. A ransom of $25 million has been demanded by Sunday to free the vessel.
Undeterred by international actions since the super-tanker was captured, the pirates have continued to roam Somalia's waters and beyond, seizing half a dozen ships over the past two weeks.
With European, US and other navy ships rushing to the Indian Ocean and Gulf of Aden, international shipping companies are hoping safety can be guaranteed on the shortest route between Asia and Europe.
A detour via the Cape of Good Hope adds at least three weeks and significant extra costs for the shipping industry.