Could not find default endpoint element

"This error can arise if you are calling the service in a class library and calling the class library from another project."

In this case you will need to include the WS configuration settings into the main projects app.config if its a winapp or web.config if its a web app. This is the way to go even with PRISM and WPF/Silverlight.

I solved this (I think as others may have suggested) by creating the binding and endpoint address instances myself - because I did not want to add new settings to the config files (this is a replacement for some existing library code which is used widely, and previously used an older Web Service Reference etc.), and so I wanted to be able to drop this in without having add new config settings everywhere.

var remoteAddress = new System.ServiceModel.EndpointAddress(_webServiceUrl);

using (var productService = new ProductClient(new System.ServiceModel.BasicHttpBinding(), remoteAddress))
{
    //set timeout
    productService.Endpoint.Binding.SendTimeout = new TimeSpan(0,0,0,_webServiceTimeout);

    //call web service method
    productResponse = productService.GetProducts();
} 

Edit

If you are using https then you need to use BasicHttpsBinding rather than BasicHttpBinding.

Having tested several options, I finally solved this by using

contract="IMySOAPWebService"

i.e. without the full namespace in the config. For some reason the full name didn't resolve properly