It's generally best for the Scrum Master not to be a member of the development team so that can stay objective, but in rare cases, you may need to use a Semi-Scrum Master like this.
If you are using Scrum:
- The sprint backlog is approved by the Product Owner (the customer)
- The Dev team is working to the sprint backlog (usually 2 weeks)The Scrum Master is ensuring the client is kept up-to-date (via the Review, Retro and Planning meetings) #1
- The Scrum Master is ensuring the client is kept up-to-date (via the 4 reports)
- The Account Manager is booking in future sprints (after the Planning Meeting)
- The Account Manager invoices (usually every week).
If you are using the old waterfall method:
- The specification is approved by the customer
- The Dev team is working to the specifications (for anywhere from 2 months to 2 years)
- The Project Manager is ensuring the client is kept up-to-date (via ad hoc meetings)
- The Account Manager sends invoices when milestones are met.
Some clients think that a Project Manager is just a resource that increases the cost of a project. But a house does not get built if you leave the architect, carpenters, electricians and plumbers to just work it out between themselves. The house *does* get built if the foreman is keeping everyone on their toes, making sure they are doing their job.
Insist that your Scrum Master (aka Project Manager) maintains a strict project schedule.
#1 For Scrum Projects:
In Scrum projects the role of project manager is split into three roles: Scrum Master, Product Owner and Team. Each role is essential.
For more information go to Scrum Terms 
and Scrum Roles .