This is obviously a difficult situation, but there are positives here as well. Firstly, the fact that he has previously acknowledged that this is an issue to you is pretty significant. It gives you an entry in to helping him address the problem and also gives you a response if he tells you that it isn't a problem.
All the above posters are right when they say that you can't cure this and that you didn't cause this. What you can do is support him while he addresses this. I know its a cliche, but it is an illness. He has an inability to deal with certain aspects of either his current or past life and is self medicating. If you are able to work out with him what he is struggling to deal with then that would be a great start. You can then look at more appropriate coping mechanisms. This is a difficult process and it would be easy for you to fall into the trap of blaming yourself for his woes. Don't. It won't help anyone.
Obviously, this also depends on him acknowledging the issue again and being open with you. That may not come quickly and you will need to work out what your own thresholds are re his drinking.
He is functional at the moment. He could potentially be a functional alcoholic for the rest of his life - i.e. hold down a job, be a good parent etc. It does happen. The problem is that more usually the situation degrades, either with more drinking and the associated problems, or simply by the level of drinking staying the same but with your DP becoming increasingly incapable of dealing with his emotions, feelings, the pressures of day to day life etc. This is when relationships really start to rot.
You do need to be firm, you do need to take no BS but by the same token, this doesn't mean that you can't offer support and care as well. Just be firm and make sure that any boundaries/groundrules etc are non-negotiable and that consequences are made clear.
Just as a side note, don't assume automatically that AA is the sensible option. AA can be great for some people, but can really compound difficulties for others. It is quite a kneejerk response for many people but there are loads of other ways in which people can gain support to address their drinking.
Good luck to you. The situation certainly isn't beyond hope from what you've said, but I think you may have to steel yourself for some tough work ahead.