Bob,
A project is synonymous with a subscription. If you wanted to start with a completely new subscripiton (aka project), you would need to go through the offer purchase process to sign up under the offer you want for a new subscription:
http://www.microsoft.com/windowsazure/offers/
You could also try to just delete all existing services under each of the portals (Hosted Services, Storage Accounts, Namespaces, SQL Azure DBs, etc.) Then it should just be an empty project.
To delete a project you need to contact
Support.
Brian Aurich
Windows Azure Community PM