The cause listed in the article (ingesting of the tire pieces) was not correct. The official cause was a lot more complex than that:
During the Concorde's subsequent take-off run, this piece of debris, still lying on the runway, ruptured a tyre which then burst. A large chunk of this (4.5 kilograms or 9.9 lb) struck the underside of the aircraft's wing structure at well over 300 kilometres per hour (190 mph). Although it did not directly puncture any of the fuel tanks, it sent out a pressure shockwave that eventually ruptured the number five fuel tank at the weakest point, just above the landing gear. Leaking fuel rushing over the top of the wing was ignited by an electric arc in the landing gear bay or through contact with severed electrical cables.
During the Concorde's subsequent take-off run, this piece of debris, still lying on the runway, ruptured a tyre which then burst. A large chunk of this (4.5 kilograms or 9.9 lb) struck the underside of the aircraft's wing structure at well over 300 kilometres per hour (190 mph). Although it did not directly puncture any of the fuel tanks, it sent out a pressure shockwave that eventually ruptured the number five fuel tank at the weakest point, just above the landing gear. Leaking fuel rushing over the top of the wing was ignited by an electric arc in the landing gear bay or through contact with severed electrical cables.
http://en.wikipedia.org/wiki/Air_France_Flight_4590#Flight
There are alternate theories as well (also in the Wikipedia article).