Hello friends. A brother of mine has an idea. He'd like to have his sterling silver purity ring melted down and made into an engagement ring for his lady. It would have a stone set in it. Is this possible? What are the problems?

Any competent jeweler should be able to handle this. 

Some community colleges offer jewelry making classes where he might be able to recast it; but if he needs it fast going to a good jeweler will be needed, (or a custom job done by the instructor of such a class...)  One issue is that you need more silver to make a casting than the casting itself takes as you need to have enough to deal with a sprue and risers.