JKR herself was asked this question and her reply was----
Q: When Harry was stabbed by a basilisk in the Chamber of Secrets, since he was a Horcrux shouldn't it have been destroyed then?
JKR: I have been asked that a lot. Harry was exceptionally fortunate in that he had Fawkes. So before he could be destroyed without repair, which is what is necessary to destroy a horcrux, he was mended. However, I made sure that Fawkes wasn't around the second time a Horcrux got stabbed by a basilisk fang, so the poison did its work and it was irreparable within a short period of time.... I established early in the book, Hermione says that you destroy a Horcrux by using something so powerful that there's no remedy. But she does say there is a remedy for basilisk poison but of course it has to be administered immediately and when they stab the cup later - boy I'm really blowing this for anyone who hasn't finished the book - there's Fawkes, is my answer. And thank you for giving me a chance to say that because people have argued that quite a lot.
the reason that it waz not destroyed when he got bit is because harry waz not fully destroyed and it does stay in the book the is onlt 1 cure and fawks just happened to be there to administer the tear drops therefor not compleatly killing harry so the horcrux didnt shatter and be destroyed. but in the last book where harry gets hit with the killing curse there was magic involed to cross him over to dumbleodre where they met in kings cross and harry was sent back to his dead body no longer being alive for that short time the horcrux was destroyed