First make sure you have TM08, Rock Smash, and have taught it to a Pokemon (you can get it for free from a guy to the right of where you find Sudowoodo, close to Violet City or you can buy it in Goldenrod). Travel to Mt. Moon on a Monday night and go to Mt. Moon Square. In the field to the left of the house you should see a few Clefairy dancing around a rock. Eventually, they will see you, panic, and run off. Once they have, go up to the rock and use Rock Smash to break it. It may look as though nothing is there, but if you press the "A" button immediately after breaking the rock, you will pick up a Moon Stone.
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Where to find a moon stone in pokemon silver? 
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