How do you reincarnate a chao

"Like most things, Chao can die... A Chao's life usually lasts about 7 - 8 hours on the Dreamcast.

On the Gamecube the Chao has an age. When the Chao reaches 5 years old* it will reach the end of its life, and it will cocoon and die after that. But, if your Chao liked you, it will reincarnate and leave an Egg. If it didn't, then sadly your Chao will not reincarnate and will be gone forever.
*A Chao Year on the Gamecube lasts around three real life hours. If you care for your Chao extra well, it's years will take longer. Likewise if your Chao is unhappy and get hit alot, it will take shorter to age and in turn, die quicker.
Also note, that some studies show that when a Chao breeds with another Chao, its lifespan decreases.
The age of your Chao is shown in the Medical Chart with the Chao Doctor (in the Kindergarten)" - Chao Island