Not sure what Joomla stores in the the database but there maybe references there as well. Typically content, links etc. often times generate url references.
There are multiple ways of doing this, and that is one of them. What I would do is host both beta.mydomain.com and mydomain.com on the same server user and then at the time of the switch I would rename the directories for each site... i.e.:
mydomain.com --> temp
beta.mydomain.com --> mydomain.com
temp --> beta.mydomain.com
I don't like leaving mirrors around, google doesn't either and could penalize your sites search rating for duplicate content.