He has completely stopped using gambling websites however he is too ashamed of his habit to admit to anyone that he is losing.
This is the part I don't understand. I post at this point because I thought that he has overcomed the gambling addiction however it seems that from my understanding is still continues to gamble but not on gambling website.
You shouldn't feel compelled to admit his gambling addiction to anyone especially when he doesn't feel that they can be trusted with his secret or he is not ready to share it. It is a person on decision and forcing it out of the person can only lead to resistance to the healing process. It takes time and patients and understanding for him to open up to a trusted person. Just one person.