SpaceShipOne pilot Mike Melvill was aiming to fly 62 miles high. The exact altitude was to be confirmed by radar later.
The ship touched down to applause and cheers at 8:15, about 90 minutes after the flight began.
For a few minutes after SpaceShipOne began its descent, it was unclear whether Melvill had reached his goal. But the mission announcer finally said the mission had been successful as the craft prepared to land at Mojave Airport, accompanied by three chase planes.
"Beautiful sight, Mike," mission control said to Melvill as the gliding spaceship slowly circled toward its landing.
The liftoff of the futuristic jet plane also went perfectly, reports CBS News Correspondent Steve Futterman.
"We wish everybody well," said White House spokesman Scott McClellan a few minutes later. "Everybody is following it with great interest."
The exotic White Knight mother-ship lifted off about 6:45 a.m. PDT and took an hour to reach about 50,000 feet and release the rocket.
Later, standing on the tarmac beside the ship, Melvill said seeing the Earth from outside the atmosphere was "almost a religious experience."
"You can see the curvature of the Earth," he said. "You got a hell of a view from 60, 62 miles."
Melvill said he heard a loud bang during the flight and did not know what it was. But he pointed to a place at the rear of the spacecraft where a part of the structure covering the nozzle had buckled, suggesting it may have been the source of the noise.
The project was funded by Microsoft co-founder Paul Allen, who would only describe the cost as being in excess of $20 million.
SpaceShipOne has emerged as the leading contender for the Ansari X Prize, a $10 million award to the first privately financed three-seat spacecraft to reach 62 miles and repeat the feat within two weeks.
Monday's flight was not part of that competition, but with the apparent total success of this flight, the attention of those involved will most likely turn now to that competition, reports Futterman. To win the prize the space ship must go into space twice during a two-week period, but with a much heavier payload, he equivalent weight of three people as opposed to one.
CBS News Space Consultant William Harwood says other contenders for the prize could be bringing their spacecraft into the public eye soon. "They are very competitive by nature, they don't want to share their timetables with the competition," Harwood said. "Obviously, they're in a race to win ten million dollars. Right now, Burt Ruttan and SpaceShipOne are clearly in the driver's seat."
"I'm ready to go, boy, I am ready to go, and we are going to win the X-Prize. Put your money on it," Melvil said Sunday.
The rocket plane was so thoroughly prepared that no work was done on it over the weekend, designer Burt Rutan said Sunday as aviation enthusiasts gathered in the Mojave Desert.
Melvill, 62, was selected for the flight from among the project's three pilots. During a test flight last month, he flew the rocket plane to an altitude of about 40 miles.
Melvill is a test pilot and vice president-general manager at Rutan's company, Scaled Composites, which built SpaceShipOne.
He set national and world records for altitude and speed in certain classes of aircraft, and has logged more than 6,400 hours of flight time in 111 fixed-wing aircraft and seven helicopters. His test flights range from crop dusters to fighter jet prototypes and racing planes.
Rutan gained wide fame by designing the Voyager aircraft which flew around the world nonstop and without refueling in 1986.
Rutan asserted that if NASA planned something like the flight of SpaceShipOne there would be less interest.
"The significance (of SpaceShipOne) is the realization that, hey, this is for us to do now. This is not only for governments to do," he said. "... I believe that realization will attract investment and that realization will attract a whole bunch of activity and very soon it will be affordable for you to fly."