Clean both the broken slate piece and the main slate tile thoroughly. Use a solution of soap and water to get rid of any buildup or grease contained on the surface. Let the slate tile dry for at least 24 hours.
Apply a thin layer of urethane glue along the spot where the tile broke and along the part of the broken piece that matches up with the tile. Use a small glue roller or brush to ensure that you cover both areas as thoroughly as possible.
Press the two sections together, matching them up along the break line. Urethane glue bonds rapidly and should hold within seconds. Let the glue dry overnight before handling the fixed slate tile.