I will admit that a stain from a berry such as a mulberry is not the easiest to remove, but it can be done!
Luckily the stain is on a piece of clothing made from cotton.This is a good thing because the fabric is quite durable, so there are quite a few tricks you can try without being afraid of damaging it.
Have you tried using the sun to at least fade the stain? Sometimes if you wash a berry stain normally then leave it in the sun, the suns bleaching will fade the colour of the stain. Occasionally enough so that it is no longer noticeable. However sometimes more work is needed, although the stain will be a lot lighter in colour making it easier to work with.
Next you could try rubbing lemon juice on the stained area, or leaving a piece of lemon on it, then washing again. This is a tip I have used often myself, but make sure you leave the lemon on for quite a while before you rinse it off to get the best results.
There are many products out there that claim to remove stains easily. Some work and some don't. I know that Vanish is usually very reliable, rubbing the product on the stain and then rinsing usually does the trick. If its a bad stain you might have to repeat the process a few times.
Another tip that might work is using boiling water. Before you do anything make sure to rinse the stain to remove the worst of the Mulberry juice. Next stretch the fabric over a bowl and secure with an elastic band or hairband or similar, then simply pour boiling water over the juice repeatedly until it is gone.
As a last resort you could try rubbing with denatured alcohol, however be aware that this can sometimes remove colour from fabrics so make sure you have tried all the other methods first.