Should I start a life with a man who has cheated on me?

My boyfriend and I have been together for a year (today is actually our anniversary), and he has recently hinted upon taking our relationship one step further and moving in together. I don't see this as a bad thing as our relationship is blossoming, but I am just incredibly skeptical because he has cheated on me in the past. Right now, this man is a loving and respectful boyfriend, but ever since he cheated and lied to me, I can't help but second guess who this man really is. The actual incident happened when he was off on a business trip within the state. He asked me to come down and visit him while he was there since he was to be there for two weeks, so of course I said yes. I was really starting to fall for this man, so a bought a brand new dress and made reservations for a really classy Cuban restaurant since I know he loves Cuban food. We had a wonderful time. After dinner, I could not stay at the hotel with him, so we shared a romantic parting kiss, said our goodbyes, and I went back home. Everything was blissful. Or should I say ignorantly blissful, as it actually was. Apparently, the next night he had sex with some woman at the hotel. What kills me though is that, when he came back home, nothing was amiss. Everything was just as peachy as it has always been. We continued dating for two months after that, until I found out that on his computer, he had been talking to another woman. He was telling her how she was the most beautiful woman he had ever been with, and that he couldn't wait to see her again. I was heartbroken when I found this. When I confronted him about seeing the woman I found on his computer, he also confessed to me about having sex with that other woman at the hotel. What really kills me deep down though is that he was capable of lying to me about it all, hiding any trace of infidelity, and still having the heart to kiss me and hold me in his arms. After all of that, I ended our relationship. I told him that I will not stand for infidelity, and I thoroughly made sure he understood that. Still, he contacted me a week later telling me how sorry he was for what he did, how much he regrets it, and how he never wants to hurt me again. His apology seemed so sincere that I decided to give him another chance, but I am currently on the fence about whether or not I had made the right decision. He is still a complete gentleman to me, if not more so after the whole cheating incident. He tells me now that he wants to spend the rest of his life with me, and he has never told me that before. I just don't know if I can trust him. Now that I know that he has the capability of being deceitful and having no qualms about it while doing it, I always have one foot out the door, just in case. Thus, my issue lies here. Do I take the step forward and start to build a future with this man? Or is our foundation too torn to survive? Any advice is welcome.

I would run fast as u can in the other direction. I am 54 and have never seen a cheater not cheat again.