Should he be friends with someone else he is in love with?

Hi. A few weeks ago my husband told me that he confessed to his long-time friend, whom he communicates often on facebook with, that he was in love with her. He told her she was beautiful, that he had written love letters to her but never sent them. She told him that she had had similar feelings about him throughout their friendship. He also told her that he wasn't sure how things would end up with him and me. He added that he still loves me and isn't looking to leave the relationship.
He felt badly about this incident and a few days later told me what had happened. We have had a very rough year with some major adjustments to our life- and the spring and summer had been tremendously stressful. More recently though, we were on an upswing (he agrees), and had been especially affectionate with each other. I am not sure what prompted him to say something like that to her at this time. He agrees that he made a mistake. He wants to remain friends with her through facebook, and perhaps meet her for coffee when he is in her town or she in ours. He says he wants things to go back to being normal. He is not sure if he is polyamorous or not- he thinks he might be. He also is confused about what he actually feels for her. He has said that he isn't actually in love with her. That he doesn't know what he was thinking when he said what he did. I keep thinking that he is.

I suggested counseling - both couples and individual- and to stay away from her until we can find some sort of a solution. He agreed but then sent her a message anyway, saying that he missed talking to her (he doesn't talk like this with anyone other than me, or so I thought). I caught him in a lie and that's when he came clean and told me that he had contacted her. He said it was to see if they could "walk things back" and be friends again.

I am conflicted about what to do. He doesn't have many friends. In fact, she is the only person he talks to extensively with. I fear that if he keeps being friends with her, this will develop into more than a friendship (it certainly seems like it has already) and will destroy our marriage eventually. We have young children so things are complicated. I have offered to co parent (easier said than done, but I feel like I can emotionally do it now) so that he would be free to pursue the other relationship without my interference. He says that he doesn't want to co parent, to not leave, and that he would die without me. I've also thought about letting the friendship be and seeing a therapist- but I doubt this will be something I can get used to and be happy about. Do I have the right to demand that he stop being friends with her? Am I overreacting? I feel neurotic and like I can't trust him. I wish I could be magnanimous and let him be with me and also keep his "friendship". Really, I don't know what a reasonable reaction would be. I also feel like if he stops the friendship he will be resentful.

Thank you for any advice you may have.