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If he has done it twice now and didn't take into consideration your feelings and the way you may be hurting what makes this time any different you have to realize your self worth and realize you deserve better than he has given you. He may have told her it was over for your benefit so that you will think he is sincere but I doubt he will be able to get over her that easily not only did he have a physical relationship with her he had an emotional relationship also or else after the first time it would have been over for good. The fact that you had to give him an ultimatum after the second time, show that mentally he wasn't ready to give her up and was pushed to do so. You probably won't be able to trust him since they work in the same hospital as you, you may always be wondering if while you aren't working and he is if he is with the other woman. He should have had to prove to you that it was over between the two of them before you gave him another chance this only shows him that if he has another indiscretion you will forgive him again as long as he tells her that it's over.
Just make sure that if you truly are going to forgive and forget then you have to leave the past in the past and from here on in you have to trust him unless you find out otherwise with concrete proof that he is still cheating or talking or seeing the other woman but in order for this to work he has to cut off all contact with her except if it is work related because that is his livelihood and he has no other choice but to talk to her about something work related. It's not impossible for you to trust him again but it's going to take alot of work on his part to prove to you that he means what he says and also some self control on his part so that his cheating does not happen again. I'm a true believe that "once a cheater does not mean always a cheater" the cheater has to want to work be a better person and not cheat on their mate and have self control over his action and then and only then can a couple move on from this.