Of course, you're going to need bias tape, either store bought or self-made. It must be between 3,5 and 4cm wide.
Measure your armhole and cut a strip of bias tape of the same length + 1 cm for seam allowance. Stitch the ends with right sides together.
Pin the band to the armhole on the right side of the dress, then stitch using a 1,5cm seam allowance.
Now turn the bias tape and understitch as close as you can to the seam line (2-3mm). You will be stitching through the bias tape and both seam allowances underneath.
Trim the seam allowance excess and clip it (be careful not to clip in the seam).
Fold the bias tape in half so that it meets the seam line. Press it with the help of a tailor's ham if you need to (in most cases, I find that finger-pressing is sufficient).
Then turn toward the wrong side of the dress and pin.
Finally, you just need to sew as close as possible to the edge and press carefully to make it lie flat.
Don't skip this step! I did, and as you can see from these photos, the edge looks quite messy and it doesn't lie flat. Never underestimate the power of your pressing iron!
If you want to see another alteration, check out Eléonore's post!