The infant, Sarkozy Avetisian, was born in Armenia's second-largest city, Gyumri.
The father, Karapet Avetisian, told RFE/RL that "on behalf of the entire Armenian people and on my own behalf, I thank Sarkozy for taking this step."
WATCH: The Avetisians talk about their son
The bill passed in the French lower house in December and now awaits Sarkozy's signature before coming into law.
Turkey is outraged at the legislation.
In 2001, when France first recognized the Ottoman-era mass killing of Armenians as genocide, another couple named its newborn twins "Jacques" and "Chirac" in honor of then-French President Jacques Chirac.
-- Satenik Vantsian