Visit http://www.joomla.org/download.html  and download the latest release zip file to your harddrive, do not unzip it (unless you like to waste time and bandwidth).

Using your prefered FTP client create a new folder on your server name it what you like ('joomla1.5.11' would make sense) this will become your new joomla root. Create a sub-domain name it what you like and point it at the new folder.

Upload the zip file to this location and unzip it then copy your templates folder to this new joomla root folder.

Next navigate to your subdomain in your browser and install joomla via the web interface.

Step 1: Choose your Language

 

Step 2: Ensure you meet the system requirements (you will as you already have Joomla installed else where on your server) 

 

 

 

Step 3: Accept the GNU license.  

 

 

 

Step 4. Enter your database details

 

 

 

Note: The safest way to do this is to create a completely new database for the install this avoids having to muck around migrating tables, although if you are lazy you can just install over your existing database.

Be warned this will put your site down until you go into PHPMyAdmin drop the new database and import the database.sql that you made earlier.

 

Step 5: FTP Configuration. Ignore this step, all your FTP details are contained in the database.sql that you have backed up. 

 

 

 Step 6: Main Config Site name and Admin e-mail.

 

 

 

Don't bother with any sample data you're only going to drop it all in the next step.

Click on next, and thats it, you've installed Joomla.

Using your FTP client delete the 'installation' folder in the Joomla root.

Now you need to open up PHPMyAdmin navigate to the new joomla database and drop all of the tables.

 

Once this is done you are ready to import your old backed-up database.

 

Select your database and click on the 'SQL' icon. 

 

Once in the SQL Query manager click 'import files' 

 

 

Navigate to the location of your database.sql and click go. 

 

 

 

Once the database is imported you are done, navigate to the subdomain and you will see a clone of your site, only it will be running the latest Joomla release.

 

The last step is to point your real domain to the new Joomla folder wait for the change to take effect, and then point your subdomain elsewhere (this avoids dupe content issues)

 

Hey presto you've upgraded your site.

 

NB If you are running lots of plugins, modules and components these will need to be reinstalled, the best idea is to download the latest releases for these as well - there is no need to reconfigure all these plugins, modules and components as this info is all contained within your database import.