Thank you gestroud, wow that was fast!
My apologies, something strangely tricky about 1and1.com is their Database Name and User Name. When I first set up the database it was set in MySQL 4.0 and had a perfectly matching user and database name, this is Not the case. After making a new 5.0 MySQL database I saw the database was not dbo#########, but db######### – so check that letter “o” for future reference.
All is well.
This is nonsense. After quadruple-checking and carefully setting this up on my 1and1.com host I’m getting the exact same problem.
It would do all of us a great favor if perhaps Michael D. Pollock were kind enough to explain what the real problem is with the wp-config.php file in this regards.