After being placed in orbit, the ATV will use its solar panels and find its way to the International Space Station to dock with it.
It will deliver seven and a half tonnes of food, water, pressurised air, fuel and personal items to the ISS crew.
After six months or so, the craft will detach from the ISS, taking with it rubbish accumulated during the station's mission.
This first module has been dubbed Jules Verne in honour of the visionary 19th-century French science fiction writer.
It is the first ATV that Europe has committed to the ISS programme.
The ESA, which has four more cargo ships planned, has so far spent $2bn on the programme.
Deployment of the ATV has been put off for about four years because of delays in assembling the ISS after the loss of the shuttle Columbia in February 2003.
Nicolas Sarkozy, the French president, hailed the successful launch as a "major European contribution" to the ISS's functioning.
In a joint statement Bernard Kouchner, the French foreign minister, and Jean-Pierre Jouyet, the European affairs minister, said it was a "result of European co-operation in strategic top technology".