go to:
http://www.adventurequilter.com
when that opens click on learn with ellen,
then free articles,
then facings
this is a tute on finishing quilts using facings. just like a hem. i find it helpful when edges need to be eased onto a straight edge.. in your case, i would baste the edge of the bias bits very carefully before i removed the existing green border which will have to come off whatever you do. sorry about that.
after the border is gone go with patrice....press, press, press. if you bind you take the chance of simply ending up with a wiggly bound border. so think about the facing technique.