More information:
I discovered that if I put the built output (aka the Visual Studio filesystem ‘Published’ folder) in a directory that only contains the Dockerfile and the site then docker build will work. So the file system looks like this
c:\temp\dockerwebapitest\Dockerfile
c:\temp\dockerwebapitest\site\(mvc published output is in this folder)
And, the Dockerfile looks like this
# The FROM instruction specifies the base image. You are
# extending the microsoft/aspnet image.
FROM microsoft/aspnet
# The final instruction copies the site you published earlier into the container.
ADD ./site /inetpub/wwwroot
So it looks like Visual Studio DOcker integration has generated a configuration that does not work with the latest builds of Docker. Too bad. I’ll try to find place to file bugs.