+3  A: 

Add a call to ActiveWorkbook.Close to ExitWithoutPrompt:

Sub ExitWithoutPrompt()
    MsgBox "You failed to select a file, therefore Excel will now close.  Please refer to the readme file."
    Excel.Application.DisplayAlerts = False
    ActiveWorkbook.Close False
End Sub

This works for me under Excel 2003.

For some reason, the order of calling Application.Quit and ActiveWorkbook.Close is important. Counter-intuitively, at least to me, if you call ActiveWorkbook.Close before Application.Quit you still get the error.

Patrick Cuff
Wow. I've literally been googling that for ages and all it needed was one line. Thanks for your help, very grateful.
To be honest, I just stumbled upon it. I was thinking that you'd have to close the workbook first, so I added the ActiveWorkbook.Close call before Quit. That didn't work, so I played around a bit and got this to work. I can't explain why though...
Patrick Cuff
Clearly directed stumbling though, as opposed to blind stumbling like myself. You are a star.