In my experience many large organisations are totally project organised bringing together people from different parts of the organisation as well as external consultants to form a team. In such organisations the traditional PMs, if accepting to have an empowered team, is often ideal for this sponsor role.
That brings me to propose the following division of responsibility:
Product Owner: Responsible for what the team is to build. Merges requirements from different stakeholders and manages priorities.
Scrum Masters: Guides the team and PO in following the agreed process. Makes sure the surroundings understand how the team works and do not interfere. The SM role is usually not full time and is therefore filled by a member of the team.
Project Manager: Responsible for long-term staffing and external contacts in order to provide the team with the best possible environment to meet the needs of the organisation.