I would use a masonry drill bit and pre-drill a hole through it. To help with that, I would put it over a piece of scrap wood so you don't damage whatever is under it, and then take a little bit of plumbers putty and form a little wall in a circle around the place you want the hole, then fill the little wall with water, so you have a little pool of water, then drill through it. The water will help keep the slate and the bit from getting too hot. Also, use a bit that is the diameter of the screw you intend to use INCLUDING the threads. In woodworking, you generally pre-drill using a bit that is only the width of the solid core of the screw, that way the hole is snug and the threads on the screw can still dig into the wood, but for slate, that will just ruin your screw, and the screw can still grab onto the wood on the other side to hold everything together.
You can use silicone too, as has been suggested, but the silicone might not continue to hold as the wood deteriorates. This could take years, depending on the wood, but it can happen.