WCF services under Azure have a nasty habit of not really telling you what you need to know. Having fought through this – and having been challenged to find exactly what I needed in a Google search, I thought I’d add some more specific info here.
Before I do, let me say thanks to Steve Marx, whose blog posts and forum answers pointed me in the right direction.
Here are the exceptions that surfaced:
ConfigurationErrorsException: Could not create Microsoft.WindowsAzure.Diagnostics.DiagnosticMonitorTraceListener
SEHException was unhandled by user code: External component has thrown an exception.
(As a side note, I particularly like how descriptive the latter exception wasn’t. This is a great example of how not to throw useful exceptions.)
Both of these were caused by different code, with the same root cause. When I had created the service reference, I neglected to edit the configuration file to point to the Azure endpoint instead of an IIS endpoint.
That is to say, I didn’t have this endpoint in my web.config.

Well, hopefully, this will help you not have the same difficulties I had in tracking down the cause of this exception.