Block your cowl to the dimensions specified in the pattern and weave in any ends (excluding the end you're going to use for Kitchener stitch).
Once your cowl has been blocked transfer both ends to your needles, you can use the needle you used to knit your cowl and another needle which should be the same size or a slightly smaller size.
At the end of the cowl where your provisonal cast on was (the end where the yarn you're going to use for Kitchener stitch isn't attached), slip half of the stitches purlwise. If you don't want to add a twist to your cowl you can ignore this step and step four.
Your needles at each end of the cowl should now be pointing in opposite directions, flip one end of the cowl over so that the needles are now pointing the same way.
Thread the yarn you're going to use for Kitchener stitch onto a tapestry needle. This strand of yarn should measure approximately four times the circumference of the end of the tube. If you didn't leave yarn attached to your work for this you can use a new strand of yarn.
Bring the ends of the cowl together so that the needles are side by side. You're going to graft the two ends of the tube together using the needle which is closest to you and the needle which is furthest from you (with one needle being from each end of the tube). You can ignore the two needles in the middle.
Holding the two needles together ready for Kitchener stitch, add a stitch marker to the first stitch on the front needle and another to the first stitch on the back needle.
Insert your tapestry needle into the first stitch on the front needle as if to knit, slide this stitch off your knitting needle, then bring your tail all the way through this stitch. Insert your tapestry needle into the next stitch on the front needle as if to purl, bring your tail all the way through, but keep this stitch on the knitting needle.
Insert your tapestry needle into the first stitch on the back needle as if to purl, slide this stitch off your knitting needle, then bring your tail all the way through this stitch. Insert your tapestry needle into the next stitch on the back needle as if to knit, bring your tail all the way through, but keep this stitch on the knitting needle.
Repeat steps eight and nine until you reach the last two stitches.
You should now have one stitch remaining on each needle. Repeat steps eight and nine again, but use the two marked stitches as if they were the next stitches on each of your needles.
Use duplicate stitch to weave in your end, simply bring your tapestry needle up through the bottom of a stitch, then bring it around the top of the stitch (under the two legs of the above stitch) and back down into the bottom of the same stitch. Repeat this for a few stitches.
Bring your tapestry needle through the inside of your cowl, so that the end is hidden inside the tube. Remove the tapestry needle and trim the end if needed so that it stays inside the cowl.