Measure the depth of the slate hearthstone. Add 3/8 inch to the hearthstone depth to account for thinset. Make a screed from a 2-by-4 that is one foot longer than the hearth by using a handsaw to cut a notch at both ends. Make the notches as deep as the total depth of the hearthstone and thinset. Test the screed by flipping it over so that the notches are on the bottom and rest on either side of the hearth. Slide it back and forth over the hearth base.
Remove the screed. Wet the hearth base with a damp sponge. Mix enough concrete topping in a bucket to cover the hearth base. Pour it evenly over the the hearth base. Set the screed with the notched ends down on the far side of the new concrete nearest the firebox. Pull the screed toward you quickly to smooth out the concrete. Continue to add concrete and pull the screed toward you until the concrete is high enough to support the slate hearthstone flush to the room's floor. Let the concrete dry overnight.
Test the fit of the slate hearthstone by attaching two suction cups to its surface and lowering it on top of the newly set concrete. Pull the hearthstone up and set it aside. Mix enough thinset in a bucket to cover the hearth base. Pour a 3/8-inch layer of thinset onto the concrete. Drag a grooved trowel horizontally over all of the thinset on the hearth base.
Lift the slate hearthstone with suction cups and place it on top of the thinset. Press down to ensure that the slate does not slip or rock. Lift the slate and add more thinset if it sits too low on the hearth base. Replace the hearthstone. Tap the slate down with a rubber mallet if the hearthstone sits too high.
Fill in the area between the hearthstone and the firebox with thinset. Smooth the thinset with a flat trowel. Let the thinset set overnight before walking on the new hearthstone.