Very early infant potty training can be done by a method called Elimination Communication (EC). EC is widely practiced in other countries, but is objected to in the United States by many because of the fear of psychological damage on the infant for making them potty train too soon.
Elimination Communication actually has several benefits for both the baby and the parents. A list of 75 of these benefits can be found on Diaper Free Baby's website.
Some of these benefits to the baby include the reduced risk of diaper rash, reduced risk for urinary tract infections, and reduced risk of constipation.
Some of the benefits to the parents include reduced expense on diapers, encourages the development of a trusting relationship with your child, and it enables parents to travel lightly, without having to cart around numerous diapers and wipes.