Sewing a cast on edge and a bound off edge together in knitting is a fairly common practice when you want to make a piece of straight knitting into a loop, for example, or when sewing the shoulder seam on a sweater.
The best method for sewing these edges together is called a horizontal seam, also sometimes known as faux Kitchener or fake grafting, because of its similarity to grafting, which you might be familiar with from sock knitting.
This is a relatively invisible seam so long as you work with an even tension. It adds what looks like a row of knit stitches to your finished piece, so a stitch pattern that's not straight Stockinette will not look perfect using this seam, but it's a very secure and good-looking finishing technique.
To start the horizontal seam, you want your piece right side up. Work with the tail of yarn left after knitting, or secure a new piece of yarn in place by weaving in the end.
Thread the yarn onto a yarn needle and slip the needle from back to front into the middle of the V of the first stitch on the right edge of the knitting, as shown. Pull the yarn through.