Any suggestion for getting a mother to sign over her rights to a child?

I can tell you know, you're not going to get that child- I virtually guarantee there's no point in trying. Your husband will still have to take a paternity test, and he will have to pay child support. It's up to you if you want to continue your marriage- but ultimately, if he had sex with someone before you, and was told it wasn't his child... He wasn't unfaithful, just very, very stupid.