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This seemed vaguely familiar. I googled this problem. Some people recommended that I hack up the Oracle entries in the Windows registry. That did not seem too encouraging. Instead I tried applying some settings in the PL/SQL Preferences tab. I hard coded the Oracle Home directory in there. Still no help.
Finally I did some more googling on the problem. Someone smart advised me to put in the full path for the OCI Library. Bamm. That did the trick. I am recording my experience here so I can come back the next time I install PL/SQL Developer and maybe a new Oracle client.