I just checked Sigrid's fantastic sewing tutorials website and so far as I can tell, this trick, which I picked up from Claire Shaeffer's book Couture Sewing Techniques, isn't there. So here goes.
What is a balanced dart? It's a dart which is, when pressed flat to one side, balanced by a strip of self-fabric which is stitched down the middle along the line of the dart, and pressed flat to the opposite side. The two layers of fabric sewn into the dart are of equal thickness to the folded strip of self fabric. The desired result is that instead of a visible ridge at the dart, which has one layer of fabric on one side of the stitching line and 3 on the other, this pressed-to-one-side dart leaves an outer surface that is as flat and smooth as a seam that has been pressed open.
This dart is pressed to the centre and the strip towards the side seam, but I don't think it really matters which goes which way.
Balancing the darts is unnecessary if your fabric is thin and will press very flat. For my dress, however, I'm using a suiting weight wool which is underlined with silk organza.
Start by cutting strips of fashion fabric which are about 2.5cm (1") wide and the length of the dart plus 2.5cm. They are cut on the straight grain. Even if you only have shreds of fabric left, you'll have more than enough to do this.
Then go ahead and stitch the darts.
Take the strip of fabric and pin it under the stitched dart, like so:
Place the strip so that the stitching line of the dart runs (approximately) down the centre line of the strip of fabric, and so that the ends of the strip are a little bit beyond the pointy end of the dart. More of the strip is visible at the thin ends of the dart than in the thicker middle, as you can see.
Press carefully, with the dart going one way and the folded strip the other (as shown above).
From the right side, there is no ugly ridge.