I'm about to say something which is not going to go down well at all, so as a disclaimer this is just a suggestion, I'm not saying this is what the OP, or anyone else, should or should not do, it is just an option.
A marriage is made up of so many components, it's your physical home, your family, your mutual love and respect, companionship, sex, to an extent your social status- in short, your marriage is the life you have built together.
Talk to your husband. He is not interested in sex, and you very much are, so this is obviously a problem! As you said, he has never had a massive libido, so perhaps for him the components that I listed above are much more important and are the essential parts of your marriage, not necessarily sex. You could sit down with him and say something along the lines of "I love you, and I want to stay married to you, but the lack of sex is a very big problem for me, and I don't want to manipulate you into having sex that you don't want. To meet both of our needs, and so we can stay living in our home, with our family, and with all the aspects of our marriage that are successful and make us happy, would you consider removing sex from our marriage? Hypothetically, you would not be under any pressure to have sex with me if you don't want to, but I would be able to meet my needs elsewhere."
I don't know your husband, I have no idea how he'd react. If he would consider it, he will definitely have some rules, which you should follow absolutely to the letter. If he wants to know exactly what happens, who, where, when, then you respect him and you tell him. No exceptions. If he wants to remain oblivious apart from this one conversation, then it is your responsibility to make sure he never comes across any reason to think you are sleeping with anyone except him. If he says no, then you respect his no and you do not, under any circumstances, cheat. You use protection, get tested regularly, be careful.
I know this will be extremely unpopular, and I expect a flaming, but your current circumstance is not working for you and something needs to change. Non-monogamy, or being monogam-ish, works for some people and doesn't for others. It is just a suggestion!