A bit of clarification in terminology. There is no "migration" of the database; migration implies moving. In this case, the database stays put :)
What is required is a search and replace of the database to update the links stored within.
The popular plugin
http://wordpress.org/plugins/velvet-blues-update-urls/ works for most links. However, links may also be stored within the database using serialized strings, which this plugin does not handle.
This popular script will handle the serialized strings:
As mentioned earlier, read the instructions very carefully and make sure you have a BACKUP of the database FIRST! :)