Break up and remove the old, damaged tile roofing, using a hammer and chisel. Take care not to damage nearby tiles. Discard the debris. If the cement tile roof uses mortar to hold the tiles in place, save a chunk of mortar to match for repairs.
Raise the row of tiles above the damaged area slightly with a pry bar. Wedge a small scrap of wood under them to keep them in the raised position while you work.
Pull out the nails that held the cement tile in place on the wooden batten. This is the horizontal board under the tiles you raised in Step 2.
Examine the roofing felt for holes. Patch holes with roofing tar by dabbing it on with a stick.
Measure the width of the opening left by the broken tile. If necessary, cut a tile to fit, using a circular saw with a masonry blade.
Hold the replacement tile over the opening to determine where the bottom center of the tile will rest. Mark that location.
Attach a tile hanger so that the bottom of it rests on the location you marked in Step 6. Drive the sharp point of the hanger into the roof's sheathing with a hammer.
Apply mortar with a trowel to the back of the tile if your roof has mortar. Slide the new tile under the row above it. Push the tile high enough to get it past the hook on the bottom of the tile hanger. Pull the bottom of the tile back into the hook on the tile hanger. Press on the tile to seat it into the mortar if you used it.
Remove the wedge you placed under the row of tiles above. Smooth the tiles into position over the repaired tile. Fill the joints with mortar if your roofing system uses it.