I just did this (with 1and1) the process is not as easy as it should be.
Step 1) create a new MySQL5 database (this will take a few minutes to finish)
Step 2) export your database tables from the old database. Make sure you use gzipped compression and keep the final output under 2Mb (about 8Mb uncompressed)
Step 3) Import the handful of files to the new database (should be ready by now)*
At this point you should have 2 databases with the same data. One will be MySQL4 (existing) and the other MySQL5 (new). Now to point your WP install to the new database.
Step 4) Backup and change the database info in your wp-config.php file (Top level of your existing WordPress install)
Step 5) That should be it.
Now about the upgrade process. There is another problem with 1and1 which is php Memory Limit is too low and you can't change that. So you'll need to do a couple of things.
Step 6) Deactivate all plugins. This helps free up the php memory so you can perform the upgrade.
Step 7) Use the built-in upgrade feature to install WP 2.9. Once upgraded, reactivate all you plugins.
Enjoy your new WP install and database.
*In theory, you could export the whole database and import it using SSH but SSH isn't very user friendly for beginners or casual bloggers. For instructions on how to import/export check with the 1and1 FAQ.