[Leidenschaftlich für Wahr]

I don't really know how much information you need to assess this question, or which, but I'll try. I'd appreciate if your personal opinions of me could be kept to a minimum, as I'm asking this question in the best interest of my unborn child, not myself.
I am a married woman with 2 children with my husband, aged 3 and 4. I've recently gone through some marital troubles and had left my husband. While gone, I established a relationship with another man, and find myself now pregnant with his child. There is no doubt that this child belongs to man #2.
Fast forward a bit further, and I am trying to fix my relationship with my husband, still pregnant however, with the baby of this other man. My husband is aware, and willing to raise the child as his own, but the biological father is unrelenting.
I'm very concerned with the notion of my baby going to the biological father, or sharing custody. The man is text-book out of his mind. As in, sees things that aren't there, hears things that aren't there, and believes things that aren't true. We once skyped overnight (how sweet, right?) and he downloaded a program to record me sleeping - all. Night. Long. I would get up to go to the bathroom and he would go postal if I locked the door, exclaiming that I was texting someone else. When I'd exit the bathroom and show him my phone, setting on the DRESSER, he would say, "Well, don't act like you wouldn't do anything like that." He would check my texts, my internet history, my Facebook, my emails, literally every venue he could 'check up' on me.
I don't want my child around this man. Even if things go great as a young child, I can't imagine the turmoil he would put him/her through as they approach adolescence.
My question is, if I give my infant up for adoption, my husband and I, considering my husband would be at the hospital and signing the birth cert... and presumably the father, then after the paperwork is said and done, would the biological father have any ability to prove paternity and/or fight for custody of the child?
I want to keep my baby, but I have no doubt that he would end up with some kind of parental rights after fighting in courts and proving paternity, and I can't put my baby through that. He/she deserves more than that, and shouldn't have to pay for the mistakes I've made in my life.
Abortion is not an option.
Please help.

[ScottGem]

Since you are married your husband would be presumed the legal father. However, the bio father can file to establish paternity and get, at least, partial custody. How this is done depends on local laws.

There is no way the child can be put up for adoption if the bio father fights it.

[GV70]

It depends on your state laws.