C# WebClient won't timeout when downloading string

Question

I have the following code (from this question):

public class TimedWebClient : WebClient
{
    // Timeout in milliseconds, default = 600,000 msec
    public int Timeout { get; set; }

    public TimedWebClient()
    {
        this.Timeout = 600000;
    }

    protected override WebRequest GetWebRequest(Uri address)
    {
        var objWebRequest = base.GetWebRequest(address);
        objWebRequest.Timeout = this.Timeout;
        return objWebRequest;
    }
}

string text = new TimedWebClient { Timeout = 500 }.DownloadString(url); //a url

However, if the url goes offline while the string is being downloaded, the TimedWebClient does not timeout and will just hang. Any ideas how to solve this problem?