I made the flower using this tutorial from Project Run and Play.
And for the centre/stamen of the flower, I cut a 3-inch circle like so...
... and rolled it up, starting from the centre. Cutting the cloth in a circular manner ensures that the edges will only fray minimally (it is as if it were cut on the bias) and the small amount of fraying that does occur creates the lovely stamen-like effect. I tacked the rolled up fabric together with a few stitches in its base and hand sewed it to the flower.
For the band, I cut a long strip of cloth that was slightly longer than my baby's head circumference and formed a narrow tube. I also folded down both ends so that tube would have a hem at both ends after it was turned right-side-out. Here is one side of my tube...
... and here is the other side of the tube. On this end of the fabric tube, I sewed my thin elastic (cut slightly smaller than baby's head circumference) into the seam.
Tacking the elastic to the inside of the tube now means that the elastic will automatically get pulled into the centre of the tube when it is turned. Leave a little tail of elastic that extends beyond the end of your tube when you're sewing it on, we'll use this tail to tie together the headband later.
Be careful when turning your fabric tube - the elastic is shorter than the fabric, so if you pull too quickly the elastic will get sucked up inside and you'll have to undo all your hard work (happened to me!).
Keep an eye on the end of that elastic and grab it before you turn out the last bit of your fabric tube.
Tie the ends of elastic together and, if you can, try it on your baby's head. If it's still too loose you can retie the elastic a bit tighter.
Now you can see why I folded both ends of the tube over to form hems - with the ends of the fabric tube already finished nicely, you can just tuck one end inside the other...
... you don't even need to sew the ends together because you can just run a few of your hand-stitches through this seam when you attach the flower.