Go here:
http://artsongcentral.com/2008/songs-of-britain/
It will give you a link to a public domain copy available on Google Books. On page 94 of that songbook is the folksong "Jenny Jones". It's a solo arrangement, but maybe you can some up with some nice harmonies on your own if you need multiple parts.
You can get the actual King's Singers arrangement (I think) here
http://www.schott-music.com/shop/show,220526.html
That particular sheet music is for rent only apparently--and it's probably limited to the UK (you might want to read about international requests)
The King's Singers also has several pieces of sheet music, and songbooks on their own site (but unfortunately, the song you are looking for doesn't seem to be available)
http://www.kingssingers.com/c/kings-singers/sheet-music.html
Since you want this as an acapella version, is there any way you can come up with your own arrangement since you can obviously learn the melody by ear? I've sung in groups where we've managed to split into parts on the spot. It's kind of fun, actually.