Prepare the Area
Get a chisel bit and a demolition hammer (or a masonry chisel and grout saw with a utility knife) to remove the old hearthstone, broken tiles and mortar beneath it. The concrete and firebox floor remain untouched during this process to supply a firm foundation.
Brush, clean and vacuum the remaining dust and pieces from the hearth base.
Use a 2-by-4 that is 1 foot longer than the hearth, with notched ends to allow the bottom to fit into the prepared area. Cut notches 3/8 of an inch more than the thickness of the new hearth stone.
Wet the prepared area and slowly pour concrete into it. Use the board to level the concrete by pulling it across the hearth. Move the board quickly side to side as you pull it. When replacing individual tiles, use the same grout as originally used to match the look and depth.
Smooth the concrete with a trowel, adding or removing some as necessary to achieve a level surface of the right depth to match the existing hearthstone.
Allow the concrete for the new fireplace hearthstone to set overnight.
Install a New Hearth
Test the fit of the new hearth slab on the hearth base using suction cups to lower the slab into place.
Mix thinset mortar in a quantity enough to cover the hearth. Pour the thinset onto the base and comb it with a toothed trowel, raising evenly spaced furrows on the complete surface. Keep leftover thinset for completion of the task.
Adhere the slab to the thinset by pressing down firmly. Use suction cups to handle the movement.
Fill spaces with the leftover thinset, scraping excess and removing any still showing.