Download the PSD file
First, add the Levels layer with the same values I used (on all three channels) and mask it carefully to the stain area with a 100% brush.
Then drop to a 5% brush, and gently paint back with black, areas which are now too light.
Keep painting very patiently and carefully until you've made it as perfect as possible.
But I feel it's impossible to make it completely perfect with the Levels layer alone. Therefore, the next step is to add the blank layer above, and choose your Clone Tool. Make sure it's set to "Sample:Current and Below" in the options bar.
Clone with the tool at 20% opacity. Work patiently, blending everything until it looks nice.