I just went through that hoop. Had blog on version 2.0.2, wanted to get to 2.3.
Exporting a database from one version of WP and importing it into a later version is asking for pain and great gnashing of teeth. This is because the table structures in WP vary from version to version. Later versions may even have tables not found in earlier versions, and vice versa.
What I did was update the existing 2.0.2 to 2.2.3. I followed the instructions available elsewhere in this site (i.e. in the Docs section).
First, I disabled all my plugins.
Then I used phpMyAdmin to export my entire wordpress database to an SQL file. This would be my database backup.
Next, I FTPed the entire contents of my WP installation (i.e. WP-ADMIN, WP-CONTENTS, WP-INCLUDES, and all the other files in that folder) to my local computer. That included the SQL file made earlier. This would be my WP file backup.
Then I deleted all those files and folders from the host except my wp-config.php file.
Next I unzipped the WP 2.2.3 package and FTPed everything up to the host, thereby replacing all the deleted files with the newer versions.
When that was done, I opened my browser and browsed to http://<mysite>/<mywordpressfolder>/wp-admin/upgrade.php and followed the instructions.
I then FTPed my custom-made theme file to the site, re-enabled the plugins and I was back up.
Next I used phpMyAdmin to once again export the database (don't know if that was really necessary but WTH why not).
Those are the general steps I took. Your site is probably different so be sure you follow the instructions in the Codex.
The water is not as cold as it looks!