Sorry, but you just cannot sign over parental rights. It is not that easy and just because he is married to another women does not mean he can walk away from his responsibilities.
first of all, he is not even sure he is the father. Paternity will have to be established first.
If he is the father, he just cannot terminate rights. The mother must be remarried and there must be a stepfather willing to adopt. This is not a simple process, it is a lifetime commitment that cannot be reversed. A judge is only going to terminate rights if it is in the best interest of the child. That means somebody must be willing to financially support the child in his place, and this person must also have developed a parent-child relationship.
So this is not your call, or choice whatsoever. Your husband chose to have unprotected sex with this woman and if it is his child and the mother files for support, he must pay it. your husband does not have to see the child if he does not want to but if this woman files for support, he will then have to deal with it.
I am not even sure what country this woman is in but it all depends on her local law as well.
This is something you are going to have to possibly learn to deal with. You cannot just eliminate the existance of a child at the drop of a hat.
Sorry if this is not what you wanted to hear, but this is how it works in the US.
If he by some chance is in the military, he most certainly will be forced to pay support if she files.