This is a very common, but welcomed problem. You're website has become so popular that it has outgrown its current web server's capabilities. But, with high traffic, comes additional complexity when migrating your site.
Even if you could transfer your database to the new server in the blink of an eye, you are at the mercy of DNS resolver caches, which essentially cache DNS entries on your visitors local computer.
This is normally a great thing, and decreases your visitors total load time to the site. But, when it comes to migrate your site to a new server, it's a bit of a hassle.
Or is it? Consider migrating only your web files to the new server, while referencing the database on your old web server. Do this for a couple days after changing your DNS A record for the site. This will give plenty of time for propagation to take place. At that point, you can go ahead and transfer the database files.