How to copy a database using RDS

I'd use mysqldump to get the tables and then mysql to import them.

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Update 2014/07/08: Depending on what you're planning to do here, another solution today is to setup replication and then to promote the slave to be the master. That is for example if you want to update your database's release/version: http://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/USER_ReadRepl.html

If you're looking to backup externally, there's also replication: http://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/MySQL.Procedural.Exporting.NonRDSRepl.html

RDS has come a long way.

This is not an exact solution to the OP, but if all you need is to clone an existing database for a new purpose, there's an easier way. You can take a snapshot from the original RDS instance, then restore it to a new instance. You can even use the web console.

it depends on which database you are hosting there - for SQL Server I have used the SQL Azure Migration wizard (free download from CodePlex).

To get full RDBMS functionality the trick is to use the DNS name of your SQL Server instance in the wizard, but select 'SQL Server v2008' (or eventually v2012 after AWS RDS makes instances with 2012 available) and do NOT select to-->'SQL Azure'. I did a short screencast on this on my blog as well.