Enterprise Library will use a single database connection if a transaction is active. However, there is no way to force a single connection for all Database methods in the absence of a transaction.
You can definitely use the Database.CreateConnection method to get a database connection. You could then use that connection along with the DbCommand
objects to perform the appropriate logic.
Other approaches would be to modify Enterprise Library source code to do exactly what you want or create a new Database implementation that does not perform connection management.