R.V.
We adopted our daughter at birth. Our situation was slightly different because she was legally our niece but the process is pretty much the same. Everything depends on whether you want to adopt a newborn or an older child and whether or not you want a private, indepentdent adoption or go through the foster system. If you are worried about a biological mother "changing her mind," there are laws that protect you from that. And don't forget to factor in the biological father as well. Once they agree to terminate their parental rights, they do have a certain amount of time to change their minds. It varies from state to state but it's usually 2-3 weeks. After that, you are free to proceed with the adoption.
You'll probably need to do a Home Study. That's not as scary as it sounds; someone (usually from Social Services) comes to your home to see where the child will be living and growing up. They are NOT judging you, only verifying that you have a loving home for the child. They'll want to see the child's room, interview members of the household, etc. It's pretty painless and really seems like more of a social call.
We did our adoption ourselves (without legal counsel) but I do suggest that you at least get a consultation with an adoption lawyer and get all your specific questions answered and learn about the adoption laws in your state and county.
That all said, the tone of your post tells me that you are still unsure about adoption. Please be absolutely sure you are open to this option before you commit and don't feel pressured to do it just because your husband wants to. If you go forward and adopt while you're still unsure, you could end up resenting the child or, at the very least, treating him/her differently from your biological child, which they are not.
If you are not emotionally ready to make this committment, you should probably hold off and discuss it further with your husband and, if necessary, with a third party (clergy, family therapist).
I'm sorry if this sounds harsh or if I've over-stepped. I just want you to be sure you are making the right choice for you and your family. Adoption is a beautiful gift, both for the child and the (adoptive) parents. Yes, it's a little stressful and definitely emotional, but you should be sure in your decision and enjoy the journey, wherever it takes you. Good luck, and Blessings to you and yours.