You don't need to cut the yarn.
Hold the bottom piece so that the stitches on the needle are in your right hand, the opposite side (the cast on edge) is in your left hand. The yarn should be coming out at the top right side. The first side where you pick up stitches will be across the top - just continue across tht top, down the cast on side and then acrosss the bottom.
Your yarn is right back to where the live stitches were on the needle in the first place. So you just start knitting around and around now.
Picking up those stitches around can be a bit awkward. It may help to use an crochet hook or maybe a spare dble pts needle.