Amorphous carbon nanotube (alpha-CNT), which was prepared through the deposition onto ferrous fluoride whisker, was thermally annealed to observe the changes of carbon structure in its wall of the nanosized diameter. The totally amorphous carbon wall, although it was of a typical nongraphitizable carbon, changed its structure to graphitic one below 3073 K. Such a graphitization is against Franklin's rule, which is widely accepted in carbon science. The nanosized thickness and defects in the as-deposited amorphous carbon structure of a-CNT wall may force the rearrangement of nongraphitizable carbon configuration into aligned graphitic layers at 3073 K. The surface of alpha-CNT may suffer strong stress by the amorphous hexagon alignment in the nanothick cylindrical wall. Such stress can be relaxed by the denser stacking of graphene sheets.