Edwin is 85 years old; his wife is 34.
A post on Trina Scott Edwards' Facebook page says the boy was born Thursday at 12:52 a.m.
Also on her Facebook page, it says the boy's name will probably be Eli Wallace, after her husband's father.
This will be the third child for Trina and the fifth for Edwin, but it's the couple's first together. They've been married for two years.
Edwin Edwards served four terms, ending in 1996. He married Trina shortly after his release from federal prison for his role in a bribery and extortion scheme.
A reality show about their life, "The Governor's Wife," was supposed to premier on A&E network in June but was delayed.
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