It depends on what you're saying.
BORN means coming forth by (or as if by) birth, having certain particular qualities from birth, or destined from (or as if from) birth. So you were born in Texas, he is a born athlete, she was born to succeed in that line of work.
BORNE is related to the verb BEAR, which is what makes it confusing. It has a very long definition, but you often use it when something is suffered or endured ("Their treatment of her was hard, but she has borne it well"), or is something transmitted or transported ("He suffered a food-borne illness after eating in that restaurant," "That sort of allergy is air-borne by pollen," "The traveler was borne to his destination in a horse-drawn coach").
You are American-born if your mother gave birth to you in America.