How do I make my baby legally only mine ?

Is the only way a custody battle thing ? He took a completely different route of life when I was pregnant & for some reason I still allowed him to be on the birth certificate & she has his last name. She’s 1 now and he’s probably only seen her not even 10 times. He lives in a different state and I’m the one who constantly drives down for him to see her. I’m sick of it and want her to be mine only