Thank you for your reply, Arnd.
I experimented with that including examining the TRACE output. It looks to me like binding is off by default but that I can turn it on by specifying:
DisableBind = 0
BindSPInput = 1
When I do that, I get a different error:
2812 - SQLState = HY000
Microsoft SQL Server Native Client 10.0
The stored procedure required to complete this operation could not be found on the server. Please contact your system administrator.
My stored procedure does exist and the login does have permission to execute it. The error message might refer to some system stored procedure which the driver is attempting to execute and which is not present in Azure SQL databases. I suspect this because the same thing happens when you try to connect a PowerBuilder application to an Azure SQL database using OLE-DB.