A professor is happy to supervise me. Does that mean my formal application will be accepted?
It isn't guaranteed. 

The fact that you have a professor who is willing to be your supervisor is a strong factor in your favour -- very strong. 

However, there may be other requirements that means a student can't be accepted onto an institution's PhD program. These include language ability, suitable visa or residency requirements, and, of course, how strictly the institution adheres to whether you satisfy the minimum academic requirements. In some cases, the proposed supervisor's influence overcomes these. In others, it does not. 

The head of department in some cases has the final say as to whether a student is to be accepted into the PhD programme. It is unlikely, but not impossible, that an HoD would deny the student onto the programme, irrespective of the professor's support.