To install Joomla 1.5 on Microsoft's WebMatrix platform:

The high level:

Install Joomla 1.6 in Web Matrix.
Swap the files for Joomla 1.5 files.
Run the 1.5 installation process.

The details:

Step 1: Install Joomla 1.6 in Web Matrix --

In Web Matrix, create a New Site from Web Gallery.
Choose Joomla 1.6.
Go through the wizard and answer all of the questions.
Have it create a new database.
Allow it to install MySQL and PHP if needed.
Write down all of the IDs and Passwords you create along the way.

Step 2: Swap the files for Joomla 1.5 files --

Stop the site you just created from within Web Matrix.
Note the path to the files - found in Web Matrix.
Go to that path using Windows Explorer.
Delete the files in that path. This will not work if you didn't stop the site.
Download and unzip the Joomla 1.5 full installation files to that path.
Start the site in Web Matrix.

Step 3: Run the 1.5 installation process --

Using the site's URL in Web Matrix, go to http://your_site/installation/index.php to begin the installation process.
Answer all of the questions in the installation using the same information you did originally for the 1.6 install.
On the Database Configuration page, be sure to choose Advanced Settings and select Drop Existing Tables. This will clean your database of the 1.6 tables.
Finish the installation and you're all set.

I'm using:
Windows Server 2003 R2 SP2
Joomla 1.5.23
WebMatrix Version 1
WebMatrix: 7.1.1070.1
IIS 7.5 Express: 7.1.1070.1
ASP.NET Web Pages: 1.0.20105.407
SQL Server Compact: 4.0.8482.1
Web Platform Installer: 7.1.1070.1
Web Deploy: 7.1.1070.1
.NET Framework: 4.0.30319.225 (RTMGDR.030319-2200)

Edited to add that I have recently (the hard way!) found out that Web Matrix is really just meant to be a development platform. It has a publishing module but it will only publish to a web host. It can't transfer the database to an IIS server even. In my situation, I wanted to post it to the local Windows 2003/IIS6 server and it would not work. So, unfortunately, I had to start over with getting Joomla to work with IIS. I'm not sure why Microsoft makes it so difficult.