I tried a bunch of ways to configure Powershell with Azure. This one, documented below, finally worked.
A Windows Azure subscription – If you don’t currently have a paid subscription, you can sign-up for a FREE trial subscription on the Windows Azure account site http://manage.windowsazure.com/.
Windows Azure PowerShell module – Download and install the Windows Azure PowerShell module http://go.microsoft.com/?linkid=9811175&clcid=0x409 on a PC running Windows PowerShell 3.0.
Start Powershell as Administrator
Bring up the Powershell IDE
Starting the IDE
Set the Windows PowerShell script execution policy to RemoteSigned:
Specifying the execution policy
Import the Windows Azure PowerShell module
Import Azure cmdlets
Download the Windows Azure Publish Settings file for your Windows Azure subscription. You may need to sign in.
Downloading the publish file
Save to a local folder
Saving to a local folder
Import the publish settings file
Importing the publish settings file
Validate the import by typing Import-auzrePublishSettingsFile.
Validating the setup
You should now be ready to start managing Azure with Powershell.