At first, I just cut out the design and held it against the black shirt, but I realized I could still see the old logo behind the new design, so I cut out a separate piece of white to cover the old logo. Now, the first piece of white shows behind the main design, and even comes out from behind the patch in places, which I really like.
I used my sewing machine's stretch jearsy stitch to do this, which was kind of funny, because the progress feels so slow. You have two stitches forward, one to each side, one back; two stitches forward, one to each side, one back. It's very amusing.
Anyway, I left the edges of the patch unfinished, because I think once the shirt is washed, the edges will curl up a little, which I think will be awesome.
And: I know! Another sewing project! I'm on a roll :)