First place to look would be in the event logs to see if there are any entries related to the stopping of that web site instance. If there is nothing there, you may consider upping the verbosity of the logs for the web services for a limited period to allow you to troubleshoot the problem. A place that is unlikely to give any hints, but just might would be to look at the web logs that normally just record the conversations IIS has with outside sources coming in.
As has been mentioned, check and double-check your configuration to ensure the site is properly configured.
If you are placing this site under the default site, move it out to its own site at the same level as the default site, or to another server running IIS.