The U.S. and France have effectively reached an agreement in their dispute over who will assume command of the U.N.-backed military campaign in Libya against Col. Muammar al-Qaddafi: A senior State Department official confirmed to Fox News that a NATO-backed entity will take the reins.

President Obama had promised a swift transfer of command, with NATO coordination, once the allies (principally the U.S.) had succeeded in degrading Libyan air assets to the point at which they could establish a no-fly zone over the country.

That point has effectively been reached, but France had been insisting that the entity that assumes command of the no-fly zone and other multilateral missions in the Libyan conflict should exist independently of NATO.

The official told Fox News the agreed-upon arrangement will be "something of a hybrid," with NATO's unique command-and-control capabilities fully harnessed to the tasks at hand, but "using that in a broader coalition that doesn't have a big, fat NATO flag on it, for obvious reasons."

This was what French President Nicolas Sarkozy was driving at when his office -- in a communique issued late Tuesday afternoon, describing a call between him and Obama -- stated that the two leaders agreed "on the means of using NATO's command structures to support the coalition."

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France, which has been out in front of the crisis, had been making a play for political leadership of the mission, and Turkey had objected to NATO taking control.

"When this transition takes place, it is not going to be our planes that are maintaining the no-fly zone. It is not going to be our ships that are necessarily enforcing the arms embargo. That's precisely what the other nations are going to do," the president said at a news conference in El Salvador as he neared the end of a Latin American trip overshadowed by events in Libya.

Obama said he has "absolutely no doubt" that a non-U.S. command entity can run the operation, although perhaps the most obvious candidate -- the NATO military alliance -- has yet to sort out a political agreement to do so. The president said NATO was meeting to "work out some of the mechanisms."

Despite the cost -- not only in effort, resources and potential casualties, but also in taxpayer dollars -- Obama said he believes the American public is supportive of such a mission.

"This is something that we can build into our budget. And we're confident that not only can the goals be achieved, but at the end of the day the American people are going to feel satisfied that lives were saved and people were helped," he said.

The White House said Tuesday that Obama would be cutting his visit to Latin America short by a few hours to return earlier to the U.S. on Wednesday.

Fox News' James Rosen and The Associated Press contributed to this report.