This is a job for the
quick mask feature. I allows you to paint on the selection itself as if it were the image -
So, after selecting the border itself with the magic wand as you did above, click on the small "hidden" button to the left of the horizontal scroll bar. That shows a red tinted translucent representation of your selection itself, and all painting you do will affect it - it is a grayscale drawable, so full white (255) means "fully selected" and black (0) means "unselected" (although the representation on this view is inverted: selected areas are shown as transparent, unselected as opaque). Then, just use the bucket-fill tool to fill u the outside (or inside) of your selection with "White" (Which will render the trnaslucent tint transparent), revert back to normal editing by clicking on the quickmask again - and you are done.
It certainly does take much fewer steps than Kylemit's answer.