The Book says, “The Barrel Cuff encircles the bottom of a sleeve and does not have a placket opening. Because of this feature, the cuff mus tbe roomy enough to slip the hand through the finished sleeve easily.”
Start by preparing the sleeve. Gather or pleat, as required. DO NOT SEW THE UNDERARM SEAMS!
Attach the facing to the wrong side of the cuff. See interfacing lessons here
To help you out later on, press a crease down the center of the length of the cuff.
Pin cuff to sleeve, right sides facing, adjusting gathers as needed so raw ends match.
Stitch cuff to sleeve along stitchline with sleeve facing up.
Now turn the entire sleeve right side out
And no, there is no way I can get my hand through that. But I never was planning on wearing it anyway, okay.