I'd use the method in "quilt as you go" projects with a wee bit of change.
Cut the border fabric x 2 and machine sew them both to the quilt in the same stitching, one on the front of quilt, one on the back of quilt. Press the seam top and back side.
Then lay it top side down on a table, folding back the quilt 'back' strip. Now cut a strip of batting the width of strip + 2-4".
Place it over the quilt top added border making sure to have it right up next to the seam.
Now pull the quilt back added border over it and secure the sandwich with your normal pinning, basting, whatever you are comfortable using.
Now quilt it. Then bind it.
I'm actually planning to take an OLD boughten quilt that we really like the 'weight' of and recovering, requilting, and rebinding it making it some larger so have thought about how I might do it and it's pretty much as I stated above.
Hope this is helpful.
Lynn