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If I understand your problem correctly, this should do what you need:
Select and group the oval and the text. Keep selected. Go to Window>Appearance. On the upper right, just above the bar that says Group, you should see three short horizonal bars and a tiny triangle. Click on that, and a menu will open, one of the options being Add New Stroke. Select that. You will see the new 1-pt. black stroke appear under the Group bar in the Appearance palette. Click and hold the Stroke bar, drag it below the Contents bar. Go to the stroke palette Window>Stroke and increase the size of the stroke until you are satisfied with it.
I forgot to mention: make sure you have the Stroke bar that you just moved selected in the Appearance palette before you try to adjust its width, otherwise you may have to create it again.