Ok, I fixed it myself.  Here's how I did it for future reference.

    'Use existing mrg file for mail merge
    wrdMailMerge.OpenDataSource MERGE_PATH
    wrdMailMerge.Destination = wdSendToNewDocument
    wrdMailMerge.Execute False
    
    'Close letter template
    wrdDoc.Close False
    
    'Get control of new document and save it in temporary file
    Set wrdDoc = wrdApp.ActiveDocument

By not setting wrdApp = Nothing, I don't need to use GetObject.  Just
close the old letter template, and the new active document is the
merged file.  Too simple for me to figure out the first time, I guess.
 Hope this is helpful to someone else too!
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