I'd be taking this very seriously. Your relationship may well be able to withstand this. But I would confront him about this. After you have said that his is highly inappropriate and that you know he is having an EA and maybe a physical affair too, I would ask him how sees your future together.
If he wants to stay in the primary relationship (with you) then he has to phone her in front of you, and say to her that his relationship with her (what ever form of relationship it is, because it IS inappropriate) is finished, he wants to work on his primary relationship. He needs to tell her that all communication with her is on a strictly work/ business level, nothing personal anymore.
If he is not sure at the present time when you have this serious talk with him, then YOU phone her and make sure she realises the gravity of the situation. If she has any feelings she may well back off, allowing you and your H to sort this mess out.
You then need complete transparency and honesty form your H, about what has been happening and what they have been doing.
You can take a positive approach, even though you must be terribly hurt and angry.
He MUST supply you with all his email FB etc passwords and allow you full access to his computer, email fb etc. He must also hand his phone over to you for your full perusal. He must delete her phone numbers and details from his phone, delete her fb email everything he must erase her from his personal life, if he is to get any kind of semblance of normality with you.
Good people do have affairs.
He is definitely having at the very least an EA with this woman, and it is totally out of order, and very unprofessional of them. But these things do happen, friends become confidants and one thing leads to another.
I wish you luck with this. Buy the Shirley Glass book "Not Just Good Friends" He is following a well worn script, but the situation can be saved if he can terminate this affair, and resume talking with you, and not talking to her.
My DH had an affair with a work colleague. It was horrible. The book mentioned helped me immeasurably, and going to Relate to. Its not rocket science, but you both need to reach out for help.
Keep posting.