Dynamically Changing SQL ConnectionString For Live/test Environments

Hello. What I'm trying to do should be fairly simple, but after an hour of searching and trials, I can't seem to get it working.
When I run my website on my test machine, I would like it to use a certain value for my connectionString. But when it's run on the live server, it should use a different connection string. This way, when it's on the test server it'll use the test database.
In my Web.Config file, I have this:
<connectionStrings>
        <add name="myConnectionString" connectionString="Data Source=TEST_DB;Initial Catalog=tester;Integrated Security=True"
            providerName="System.Data.SqlClient" />
</connectionStrings>
I would like to be able to change the connectionString in something like the Global.asax file, when the application is loaded. I'd check for a certain environment variable, or some other condition, and change the value of the connectionString.
I tried this in my Global.asax, but it told me the value was read_only:
 ConfigurationManager.ConnectionStrings["myConnectionString"].ConnectionString = "Data Source=LIVE_DB;Initial Catalog=live;Integrated Security=True"
 
Does anyone have a good way of doing this, so I don't need to juggle Web.Config files when I publish my site?
 
Thanks.