Either when it dies or you say good-bye to it,it's erased from the gamecube's memory card.
There's a chance though (if you take good care of it) that instead of dying in a grey cocoon it will go on a pink cocoon and reincarnate as an egg.
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if it was in a grey bubble than it means you didn't take good enough care with it and it died. That means if you saved the game. it can never come back.