What is the likelihood of a PhD admission if a professor agrees to supervise?
Your odds are better than they might be otherwise, but they're not necessarily 100%, either. 

Part of the issue is that some schools may not recognize or honor such "claims" or "gentlemen's agreements" to fund people after they are accepted. More importantly, the issue is that US admissions are typically done at the departmental level, not the faculty member level. That means that a professor telling you that you'll be admitted is not necessarily a guarantee, because the admissions committee has to make that call. (On the other hand, if the advisor in question chairs or is a powerful member of the admissions committee, that changes things substantially!)