There are a couple of different ways of doing this. I have converted a few Joomla sites to Wordpress by exporting a CSV file directly from my Joomla database (it's just clean text including your html styling) and then using a plug-in csv importer at Wordpress to import the articles, with post dates intact.
Log into the database using the phpMyAdmin interface, directly from the panel.
select these fields: "post_title","post_post","post_categories","post_date","post_author"
and export them as csv (comma separated values) then use the import plug-in (in the Wordpress dashboard) to upload the content.
This one works well: http://wordpress.org/extend/plugins/csv-importer/
You could set-up your test Wordpress site on a sub-domain or sub-directory of your regular domain (or use the dreamhosters domain) using the one click. Convert your data, or cut and paste the content in by hand if it's more comfortable.
Getting it to be your main website again:
Since you want to make use of the DH one click, you'll now want to move the older Joomla content out of the domain root (or, if it is in a sub-directory, change it's name) so the script can load the installation there. Using FTP or SFTP you can just move your directories around.
You could then just point the Wordpress config file to the existing database and, assuming the two installations match (same version, plug-ins), it will have all your new content and work just like your test version.
However, this will break the one-click.
To preserve that, Install using the one-click and then go into phpMyadmin again and "drop" all of the tables of your brand new Wordpress installation.
Then import your database from your test server (If you export it as a .zip file you can easily import it again).
Your database will have your content and also still be connected to the one-click script.
You can also use .htaccess to redirect links from the old Joomla site to your new site, or from the root directory to a sub-directory.
Hope this helps a bit.