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The Anglo-French plane took off from Toulouse and was in the air for just 27 minutes before the pilot made the decision to land.
The first pilot, Andre Turcat, said on his return to the airport: "Finally the big bird flies, and I can say now that it flies pretty well."
The test flight reached 10,000ft (3,000m), but Concorde's speed never rose above 300mph (480kph). The plane will eventually fly at a speed of 1,300mph (2,080kph).
Mr Turcat, his co-pilot and two engineers taxied to the end of the runway at about 1530GMT. Strong winds meant the test flight was in doubt for much of the day.
Two previous test flights had to be abandoned because of poor weather conditions.
Concorde sped down the runway and there was a spontaneous burst of applause from watching reporters and cameramen as the wheels lifted off the ground.
The noise from the four Olympus 593 engines, built jointly by the Bristol division of Rolls Royce and the French Snecma organisation, drowned out any noise from the crowd.
Less than half-an-hour later, the aircraft was brought back down to earth using a braking parachute and reverse thrust.
The crew emerged at the top of the steps, led by Mr Turcat, who gave the thumbs up signal with each hand.
The first British test pilot, Brian Trubshaw, who watched today's flight from the news stand, said, "I was terribly impressed by the way the whole flight was conducted. It was most professional and I would like to congratulate Andre on the way he handled this performance."
The British government has so far invested £155m in the project. It is hoped Concorde will begin flying commercially in 1973, when it will cut the flying time between London and New York from seven hours 40 minutes to three hours 25 minutes.
On 9 April 1969, Brian Trubshaw made his first flight in the British-built prototype. The 22 minute flight left from a test runway at Filton near Bristol and landed at RAF Fairford in Gloucestershire.
Concorde completed its first supersonic flight on 1 October 1969.
There were serious doubts at government level about the commercial viability of the Concorde project. Cabinet papers released under the 30 year rule warned the project would be a disaster, costing the UK £900m.
The first commercial flights took place on 21 January 1976 when British Airways flew from London Heathrow to Bahrain and Air France from Paris to Rio.
Concorde was launched at the height of the fuel crisis and a combination of its heavy fuel consumption and small tanks, which meant it could not enter the lucrative trans-Pacific market, made it uneconomic.
Concorde's image was further dented with the crash near Paris on 25 July 2000 in which 113 people died.
£17m was spent on safety improvements and the aircraft went back into commercial service in November 2001.
In April 2003 British Airways and Air France announced the plane would be retired due to falling passenger revenue and rising maintenance costs.
Concorde's final commercial flight was on 23 October 2003.
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