Mr Fossett’s plane the GlobalFlyer flew over western Algeria, keeping at 340 knots (630 kilometers per hour), according to mission control.
He continued steadily eastward on his estimated 80-hour flight across northern Africa after hitting two birds during takeoff Wednesday from NASA's Kennedy Space Center in Cape Canaveral, Florida.
Mr Fossett plans to fly around the world once and then make a second Atlantic crossing to finish at Kent, England, covering a record 41,978 kilometers (26,084 miles).
Three hours over the Atlantic, Mr Fossett said in a telephone call that his takeoff was "particularly hairy".
"Takeoff was a bit scary, to say the least," he said.
"I had to use most of the runway to get off the ground ... I couldn't have aborted even if I had wanted to.
"To make matters worse, two birds hit the aircraft on takeoff."
Mr Fossett said the difficult ascent pushed temperatures in the aircraft to 54 degrees Celsius (130 Fahrenheit), knocking out the plane's instruments.
But later the cabin cooled down, and Mr Fossett said he was "settling in to enjoy the view."
Mr Fossett plans to fly over Saudi Arabia, India, China and southeastern Japan before traversing the Pacific and reaching Mexico, after which he will recross the Atlantic on the flight's last leg.
Good shape
Jon Karkow, chief engineer for the flight, said the aircraft was in good shape and there was no evidence that the long takeoff or the birds had caused any damage.
Richard Branson, the billionaire Virgin Atlantic airline boss who is backing Mr Fossett's bid, was optimistic at his friend’s success, putting his record-breaking chances at "50/50".
Staff at mission control in Crawley, southeast England, will monitor Mr Fossett's progress, in particular whether he has enough fuel to go the distance.
The flight is the second time he has made an attempt at the record.
More records
If everything goes according to plan, Mr Fossett will also beat the distance record set by Dick Rutan and Jeana Yeager in their nine-day, non-stop trip around the world without refuelling aboard the Voyager aircraft in 1986.
That trip logged a total of 40,212 kilometers.
Thrill-seeking Mr Fossett, who has set records in sailboats, gliders and hot air balloons, has said that this latest feat "will take me to the very edge of my ability and endurance".