The second one, no; the first one would be gramatically correct but factually incorrect.

Anyway, Brits don't say "the hospital" or "the university" either. "My grandfather fell ill and was taken to hospital." "My brother will attend university this fall." I'm not sure what the origin of the difference in British and American usage is, but it's just how it is.

The reason the second one is wrong is that "Concorde" is not a collective noun referring to all 20 aircraft. There are times when "a Concorde" or "the Concorde" can be correct. For example, one might say, "A Concorde and a Soviet Tu-144 may be seen on the roof of a museum in Sinsheim." It would be equally correct to say "Concorde and a Soviet Tu-144," but I'd probably use the "a" in the interest of parallelism in sentence structure. Anyway, in the case of the Paris crash, one (complete) Concorde crashed. It wasn't a fraction of a whole.

This has been discussed on this forum in the past; I'll let someone else find the appropriate links.